THE  LIBRARY 

OF 

THE  UNIVERSITY 
OF  CALIFORNIA 

LOS  ANGELES 


SOI 

UNIVER  rORNlA 


f  "- 

_i 

LIF. 


.The  Teaching  of  Arithmetic 


John  C.  .Stone 


•, 
-at 


I 

-?t 

PREFACE 

THIS  book  is  a  discussion  of  the  aims  and  purposes  of  a 
course  in  arithmetic  and  of  the  methods  of  presenting  each 
topic  that  should  find  a  place  in  our  elementary  schools. 
/->  Hence  it  is  essentially  a  book  for  teachers,  supervisors,  and 
—  those  preparing  to  teach,  and  thus  meets  the  needs  of  a  text- 
£;  book  for  normal  schools  and  teachers'  reading  circles. 

The  successful  method  of  teaching  any  topic,  however, 

p  depends  upon  a  clear  knowledge  of  the  fundamental  prin- 

^  ciples  upon  which  the  topic  is  based.     Hence  in  the  dis- 

J^~  cussion  of  each  topic,  the  fundamental  principles  involved 

are  fuHy  presented  in  connection  with  the  method  of  the 

Classroom  presentation  of  the  topic ;   in  fact,  they  form  a 

very  essential  element  of  the  so-called  method  of  teaching. 

Several  of  the  chapters  are  summaries  of  lectures  given 

before  teachers'  associations  and  institutes  and  it  has  been 

^  the  requests  from  such  audiences  for  the  printed  lectures  that 

r^  has  encouraged  the  author  to  bring  them  all  together  in 

l<3  this  form  in  the  hope  that  they  will  have  a  wider  influence 

• —  upon  the  teaching  of  arithmetic. 

JOHN  C.  STONE. 

April,  1918. 


iii 


CONTENTS 

PAOB 

j*  GENERAL  PRINCIPLES  AND  SUGGESTIONS  ....  1 

-  THE  AIM  OF  A  COURSE  IN  ARITHMETIC  ....  8 

COUNTING  .  .  . 25, 

THE  PRIMARY  FACTS  OF  ADDITION:  WRITTEN  ADDITION  .  30 

THE  PRIMARY  FACTS  AND  PROCESSES  OF  SUBTRACTION  .  46 

THE  PRIMARY  FACTS  AND  PROCESSES  OF  MULTIPLICATION  .  52 

THE  PRIMARY  FACTS  AND  PROCESSES  OF  DIVISION  .  .  J61_ 

_.,'.-»THE  USE  OF  GAMES  IN -NUMBER  WORK  .  .  .  .70 

COMMON  FRACTIONS  . 95 

DECIMAL  FRACTIONS 117 

PERCENTAGE  AND  ITS  APPLICATIONS  .  .  .  .  .  123 

DENOMINATE  NUMBERS  AND  MEASUREMENTS  .  .  .  149 
THE  PURPOSES  AND  NATURE  OF  PROBLEMS  .  .  .  .163 

THE  ANALYSIS  AND  SOLUTION  OF  PROBLEMS  .  .  .  174 

PLANNING  THE  LESSON 191 

THE  COURSE  OF  STUDY  IN  ARITHMETIC  ....  206 

MEASURING  RESULTS  221 


THE  TEACHING  OF  ARITHMETIC 

CHAPTER  I 
GENERAL  PRINCIPLES  AND  SUGGESTIONS 

ECONOMY  AND  EFFICIENCY 

THE  most  important  problems  in  the  teaching  of  arith- 
metic are  those  of  economy  and  efficiency.  How ;  to  put 
the  child  in  possession  of  the  essential  facts  of  the  subject, 
how  to  develop  habits  that  lead  to  the  economical  and 
efficient  use  of  these  facts  in  the  real  situations  that  arise 
in  everyday  life,  and  how  to  accomplish  this  with  a 
minimum  of  waste  are  the  problems  that  confront  every 
thoughtful  teacher. 

NUMBER  WORK  BEGUN  Too  EARLY 

Pestalozzi,  in  Europe,  and  Colburn,  in  our  own  country, 
are  recognized  as  the  leaders  of  the  movement  to  bring 
instruction  in  arithmetic  down  into  the  very  lowest 
grades  and  within  the  grasp  of  children.  As  an  excuse 
for  this,  Colburn  says,  in  the  preface  of  his  First  Lessons : 
"  As  soon  as  children  have  the  idea  of  more  or  less,  and 
the  names  of  a  few  of  the  first  numbers,  they  are  able  to 
make  small  calculations.  And  this  we  see  them  do  every 
day  about  their  playthings,  and  about  the  little  affairs 

1 


2  THE   TEACHING  OF  ARITHMETIC 

which  they  are  called  upon  to  attend  to.  *  *  *  To 
succeed  in  this  (the  science  of  numbers),  however,  it  is 
necessary  rather  to  furnish  occasions  for  them  to  exercise 
their  own  skill  in  performing  examples,  rather  than  to  give 
them  rules.  *  *  *  By  following  this  mode,  and  making 
the  examples  gradually  increase  in  difficulty,  experience 
proves  that  at  an  early  age  children  may  be  taught  a 
great  variety  of  most  useful  combinations  of  numbers." 
All  observers  of  children  will  agree  with  Colburn  that  they 
are  able  to  make  simple  calculations  about  their  play- 
things and  about  their  own  little  affairs.  The  work, 
however,  should  stop  here  until  wider  experiences  and 
further  needs  demand  greater  knowledge  of  numbers. 

iln  the  past  we  have  begun  number  work  entirely  too 
early  and  have  forced  adult  applications  of  number  be- 
fore the  needs,  interests,  and  experiences  of  the  child 
Vwere  ready  for  them.  We  have  tried  to  force  facts  and 
processes  not  needed  in  the  natural  activities  of  child- 
hood, either  in  or  out  of  school.  Thus  we  have  violated 
every  principle  of  economy  in  teaching  in  not  finding  the 
child's  needs  and  interests  and  teaching  the  right  thing 
at  the  right  time. 

THE  CHILD'S  NEED  OF  NUMBER 

The  child  has  certain  needs  of  number  during  the  first 
year  of  his  school  life  that  should  be  met.  But,  in  general, 
these  needs  are  confined  to  counting  by  ones,  tens,  and 
fives  to  one  hundred ;  to  reading  numbers  to  100  in  order 
to  find  a  page  in  his  book,  or  perhaps  to  1000  in  order  to 
find  a  house  number  if  he  lives  in  a  city;  to  the  few 


GENERAL   PRINCIPLES  AND  SUGGESTIONS       3 

simple  combinations  that  he  may  need  in  his  own  little 
affairs ;  to  the  meaning  of  halves,  thirds,  and  fourths  of 
single  objects;  to  the  Roman  numerals  to  XII,  in  order 
to  tell  time ;  and  to  those  common  units  of  measure  that 
are  needed  in  his  home  or  school  life.  He  has  no  natural 
need  of  any  of  the  written  processes  yet,  and,  in  fact,  he 
makes  figures  with  such  conscious  effort  and  knows 
automatically  so  few  facts  that  to  give  written  work 
before  the  third  grade  is  to  establish  habits  of  computa- 
tion that  must  be  overcome  later. 

EDUCATIONAL  PRINCIPLES  INVOLVED 

The  economic  mastery  of  the  subject  of  arithmetic 
demands  that  the  teacher  observe  the  principles  that 
knowledge  to  be  real  must  be  founded  upon  the  actual 
experiences  of  the  individual  learner;  that  knowledge  to 
be  retained  must  be  given  an  opportunity  for  use;  and 
that  a  necessary  condition  for  learning  is  that  the  process 
be  self-actuated  through  motive  or  interest.  Hence,  the 
subject  must  be  made  to  minister  to  the  child's  needs 
and  kept  within  his  actual  experiences.  It  is  when  we 
select  material  from  the  world  of  adults,  dealing  with 
matters  with  which  children  have  had  no  experiences, 
and  in  which  they  have  no  interest,  that  we  are  asking 
the  impossible  of  children. 

MORE  DRILL  WORK  NEEDED 

There  are  certain  essential  facts  and  processes  that 
must  be  made  automatic.  It  is  not  sufficient  that  a  child 
knows  that  7  and  8  are  15,  but  he  must  recall  automatically 


4  THE    TEACHING  OF   ARITHMETIC 

the  fact  when  seeing  the  figures  or  hearing  the  numbers 
called,  without  consciously  recalling  the  meaning  of  the 
figures  or  the  "how  many"  denoted  by  the  numbers, 
just  as  he  sees  letters  and  recalls  words  without  recalling 
the  sounds  represented  by  the  letters.  The  fundamental 
facts  are  clarified  or  rationalized  by  some  concrete  object 
or  by  a  problem  of  conditions  very  familiar  to  the  child. 
But  when  the  meaning  of  the  facts,  processes,  and  terms 
used  has  thus  been  made  clear,  next  in  order  is  drill  with 
abstract  numbers  until  these  facts  or  processes  become 
automatic.  Sight  drills  must  be  used  before  mental 
drills  so  as  to  furnish  a  mental  image  for  the  mental 
work.  Suppose  there  are  placed  upon  the  blackboard  the 
7  / 

figures  _8 .    The    child    images    this    combination    and 
15 

7 

when  later  he  sees  8_,  he  recalls  15.  Then  when  he  hears 
"seven  and  eight,"  he  has  a  mental  picture  of  these  forms 
and  can  recall  the  s,um.  Experiment  shows  that  the 
average  person  cannot  grasp  a  group  of  objects  larger 
than  four  or  five  without  analysis;  that  is,  without 
breaking  them  up  into  groups  and  adding  the  groups. 
Then  the  idea  of  a  number  as  large  as  8  -f  7  =  15  is 
not  obtained  through  objects;  that  is,  the  fact  is  not 
made  automatic  through  seeing  objects,  but  through  see- 
ing the  figures  or  from  hearing  "eight  and  seven  are 
fifteen"  until  a  mental  image  is  formed.  The  objects,  \ 
then,  are  to  make  clear  the  meaning  of  a  fact  or  process ; 
but  they  do  not  help  the  memory  to  retain  such  a  fact. 


GENERAL  PRINCIPLES  AND  SUGGESTIONS      5 

HABITUATION  rs.  RATIONALIZATION 

The  subject  of  arithmetic  is  taught  very  largely  to-day 
by  inductive  methods;  that  is,  the  child  is  led  through 
simple  concrete  illustrations  and  objective  presentations 
to  discover  his  own  facts,  rules,  and  definitions ;  but  there 
is  danger  that  this  phase  of  the  work  may  receive  undue 
emphasis.  It  frequently  happens  that  a  teacher  is  able 
to  develop  a  subject  very  clearly  and  interestingly,  and 
yet  her  work  may  lack  effectiveness,  owing  to  her  neglect 
of  drill  work  necessary  to  fix  the  facts  and  to  give  skill  in  ' 
computation. 

There  should  be  a  sharp  discrimination  between  those 
facts,  the  rationalization  of  which  is  of  vital  importance, 
and  those  in  which  it  makes  but  little,  if  any,  real  differ- 
ence as  far  as  the  efficient  use  of  the  facts  is  concerned. 
-Thus  it  is  of  vital  importance  that  a  pupil  know  the 
meaning  of  addition  and  of  such  expressions  as  "five  and 
four  are  nine,"  etc. ;  but  it  makes  little,  if  any,  difference 
whether  or  not  he  knows  why  we  "  carry."  The  important 
thing  is  that  he  has  the  proper  habit  of  carrying.  Like- 
wise, in  "long  multiplication"  he  must  have  the  habit 
of  putting  the  partial  products  where  they  belong ;  but  it 
makes  little,  if  any,  difference,  when  first  presented, 
whether  he  knows  why  they  are  so  placed. 

When  rationalization  aids  the  memory  in  retaining  a 
fact,  or  when  it  renders  the  use  of  it  more  safe  when 
applied  to  problems,  then  rationalization  should  receive 
proper  emphasis;  otherwise,  it  is  of  minor  importance. 
But  in  such  cases  as  "carrying,"  "borrowing,"  and  the 


6  THE   TEACHING  OF  ARITHMETIC 

correct  placing  of  partial  products,  the  proper  habit  is 
the  important  thing  and  rationalization  a  very  unim- 
portant thing  when  the  subject  is  first  presented.  On 
the  other  hand,  the  rationalization  of  a  rule  in  fractions 
renders  the  use  of  it  more  safe  and  aids  the  memory  in 
retaining  the  "how."  It  is  not  uncommon  to  see  gram- 
mar school  and  high  school  pupils  add  f  and  f  and  get  i5^, 
adding  numerators  and  denominators.  Proper  rationaliza- 
tion should  have  prevented  such  an  error. 

Likewise,  all  rules  for  mensuration  are  more  easily 
remembered  if  rationalized  through  an  objective  pres- 
entation. 

STANDARDS  OF  SPEED  AND  ACCURACY 

In  the  past  we  have  had  no  standards  of  skill  in  com- 
putation. Each  teacher  has  had  her  own  standards. 
Pupils  have  been  marked  and  passed  by  these  standards. 
One  teacher  has  emphasized  the  "how"  to  solve  a  prob- 
lem, regardless  of  the  time  required  to  compute  the  result 
or  the  accuracy  of  it.  Others  have  emphasized  merely 
the  power  to  compute.  Recent  tests  show  startling 
variations  in  the  work  of  pupils  from  schools  reputed  to 
be  of  the  very  highest  standing.  Such  tests  as  the 
"Courtis  tests,"  when  given  to  enough  pupils,  may,  by 
taking  a  general  average,  show  the  average  of  what 
schools  are  attaining  by  present  methods;  but  they  in 
no  way  show  what  may  be  expected  of  pupils  by  properly 
devised  drills.  From  government  statistics  we  may 
find  the  average  production  of  wheat  or  corn  for  a  state 
or  the  nation;  but  that  does  not  show  what  should  be 


GENERAL   PRINCIPLES  AND  SUGGESTIONS      7 

expected  from  improved  methods  of  cultivation  or  the 
most  economical  yield.  If  careful  records  are  kept  of 
the  progress  made  by  some  such  methods  of  teaching 
and  system  of  drills  as  are  suggested  in  this  book,  we 
may  ultimately  know  what  should  be  expected  of  a  pupil 
in  accuracy  and  rapidity  in  the  various  grades.  If  some 
other  method  shows  different  results,  we  may  thus  judge 
the  efficiency  of  the  method  used  and  thus  have  a  means 
of  judging  the  real  value  of  various  methods.  At  present, 
however,  a  method  is  considered  good  or  bad  by  a  teacher 
as  it  meets  or  fails  to  meet  the  particular  whim  of  that 
individual  teacher. 


CHAPTER  II 
THE  AIM  OF  A  COURSE  IN  ARITHMETIC 

IMPORTANCE  or  AN  AIM 

OUR  methods  of  teaching,  and  the  nature  of  the  sub- 
ject matter  that  we  teach,  depend  very  largely  upon  our 
belief  in  what  the  subject  may  do  for  the  learner;  for, 
if  we  believe  that  our  pupils  should  obtain  certain  results 
from  our  teaching,  it  naturally  follows  that  we  are  going 
to  shape  our  instruction  and  select  our  material  in  the 
way  that  seems  to  us  best  fitted  to  bring  about  those 
results. 

For  example,  if  a  teacher  considers  that  the  only  pur- 
pose of  arithmetic  is  to  train  pupils  to  compute  skillfully, 
she  will  reduce  her  instruction  to  drills  that  will  bring 
about  efficiency  in  computation.  If  problems  are  given 
at  all,  they  will  be  given  merely  in  order  to  furnish  an 
occasion  for  computation,  and  their  nature  will  not  be 
given  serious  attention.  On  the  other  hand,  if  she  con- 
siders that  the  purpose  is  "mental  discipline,"  then  she 
will  pay  less  attention  to  computation  and  to  the  me- 
chanical side  of  the  subject  and  spend  a  great  part  of  the 
time  in  developing  and  rationalizing  the  processes  and 
in  selecting  problems  having  just  the  proper  complexity 
to  task  the  child's  powers  to  discover  what  to  do.  The 

8 


THE   AIM  OF   A   COURSE   IN  ARITHMETIC      9 

problems  will  not  be  selected  for  any  real  purpose  that 
they  can  serve  in  the  child's  life,  but  they  will  be  selected 
to  furnish  a  sort  of  mental  gymnastic  exercise.  Or,  she 
may  consider  that  the  whole  purpose  of  arithmetic  is  to 
furnish  a  social  insight  into  current  business  or  industrial 
practices.  In  this  case,  she  will  devote  most  of  her  time 
to  real  problems  that  come  up  in  life,  regardless  of  the 
mathematics  employed,  and  spend  much  time  in  dis- 
cussing the  social  phases  of  situations  from  which  the 
problems  were  made,  disregarding  drills  and  the  accuracy 
of  the  solutions.  And  thus  we  might  go  on  with  one 
hypothetical  case  after  another  to  show  that  a  discussion 
of  the  aims  of  a  course  in  arithmetic  becomes  of  funda- 
mental importance  in  discussing  methods  of  teaching. 

TRADITIONAL  VALUES  OF  THE  STUDY  OF  ARITHMETIC 

The  values  of  arithmetic  have  varied  with  the  devel- 
opment of  the  race  and  have  depended  upon  the  civiliza- 
tion of  the  people  using  the  subject.  The  knowledge  of 
numbers  must  have  been  coeval  with  the  human  race. 
With  the  notion  of  distinctness  —  that  this  is  separate 
from  that  —  the  notion  of  number  must  have  had  its 
origin.  One  can  scarcely  carry  on  the  simplest  conver- 
sation or  make  any  communication  with  another  without 
a  use  of  some  knowledge  of  number.  The  savage  on 
returning  from  the  hunt  no  doubt  communicated  to  his 
family  or  the  tribe  the  amount  of  game  he  had  seen  or 
killed,  possibly  by  the  use  of  his  fingers  or  hands  or  by 
pebbles  or  shells.  To  him,  arithmetic  was  but  a  matter 
of  convenience.  But  when  he  began  to  barter  skins  or 


10  THE   TEACHING  OF   ARITHMETIC 

game  for  corn  or  trinkets  and  began  to  acquire  property, 
there  grew  up  a  necessity  for  a  knowledge  of  the  relation 
of  numbers  —  a  comparison  of  the  number  of  one  group 
with  that  of  another,  or  a  comparison  of  their  values. 
His  arithmetic  now  became  a  necessity  —  it  took  on  a 
utilitarian  value. 

As  the  race  advanced  in  civilization  and  schools  for 
the  training  of  youths  sprang  up,  arithmetic  took  on  a 
new  meaning.  It  was  given  a  disciplinary  value  as  well 
as  a  utilitarian  value.  But,  just  as  our  real  need  of 
number  has  grown  up  slowly  and  naturally  and  has  thus 
gradually  changed  the  utilitarian  values  of  arithmetic,  so 
has  our  knowledge  of  the  laws  of  mental  growth  changed 
our  thoughts  about  the  disciplinary  value  of  the  subject. 

Arithmetic  was  once  taught  to  make  one  quick-witted 
and  enable  him  to  see  a  point  quickly  and  thus  aid  him 
in  argumentation.  This  concept  of  the  value  of  arith- 
metic led  naturally  to  catch-questions  and  puzzles. 

Later  there  grew  up  under  the  teaching  of  the  so- 
called  "faculty"  psychology  the  doctrine  of  "formal 
discipline."  This  was  a  belief  in  the  transfer  value  of 
abilities  developed  by  one  study  to  all  other  lines  of 
thought,  whether  related  or  not.  This  led  to  studying 
a  subject  in  order  to  train  the  reason,  or  the  memory, 
or  the  imagination,  etc.  This  doctrine,  •  however,  did 
not  result  in  any  great  change  in  the  nature  of  the  prob- 
lems formerly  given,  but  it  merely  gave  a  new  reason  for 
retaining  the  type  of  problems  already  in  use.  But  it 
did  make  a  great  change  in  the  methods  of  teaching. 
There  was  a  change  from  the  old  dogmatic  rule  method 


THE   AIM   OF   A    COURSE   IN   ARITHMETIC      11 

to  the  demonstrated  rule  method,  and  later  to  the  heuristic 
or  development  method  of  presenting  the  facts  and 
processes.  While  a  belief  in  the  doctrine  of  "formal 
discipline"  undoubtedly  resulted  in  improved  methods 
of  teaching  over  those  methods  that  preceded  it,  it  never- 
theless placed  undue  emphasis  upon  the  rationalization 
of  the  facts  and  processes,  particularly  in  the  lower  grades, 
and  it  caused  many  obsolete  topics  and  processes  to  be 
retained  in  the  schools  long  after  they  could  be  justified 
upon  practical  or  social  grounds. 

DOCTRINE  OF  FORMAL  DISCIPLINE  DENIED 

A  few  years  ago  psychologists  began  to  doubt  the  doc- 
trine of  formal  discipline  and  to  deny  the  transfer  of  abilities 
gained  in  one  field  of  work  to  unrelated  fields.  By  many 
teachers  this  new  attitude  of  psychologists  was  entirely 
misunderstood.  Much  careless  teaching  and  indifference 
toward  the  subject  of  arithmetic  resulted.  A  teacher 
recently  said  to  the  writer  that  her  class  in  arithmetic 
was  very  poor,  and  then  added,  "But  if  they  are  poor  in 
anything,  I  suppose  that  it  had  better  be  in  arithmetic 
than  in  anything  else."  Then  she  asked,  "Hasn't  arith- 
metic been  taught  in  the  past  for  nothing  but  discipline, 
and  now  hasn't  it  been  found  that  there  is  no  such  thing  ?  " 
It  is  just  such  an  attitude  as  this,  resulting  from  a  mis- 
understanding of  recent  educational  discussion,  that  has 
seriously  affected  the  teaching  of  arithmetic. 

Our  modern  psychology  views  our  mental  life  as  made 
up  of  specialized^abilitles  «.nH  j^jijjjLgpjiargitp  ffl.rnlt.ies, 
each  trained  by  some  special  kind  of  study.  The  result 


12  THE   TEACHING  OF  ARITHMETIC 

is  that  we  now  train  for  habits  of  accuracy,  of  attention, 
of  analysis,  of  comparison^  of  judgment,  etc.,  each  in  its 
own  field.  It  follows,  then,  that  each  topic  taught  must 
have  some  bearing  upon  the  life  of  the  individual  as  a 
sort  of  socializing  factor  in  order  to  justify  its  place  in 
the  curriculum. 

THE  PRACTICAL  VALUES  OF  ARITHMETIC 

As  would  naturally  appear  from  the  discussion  of  the 
last  topic,  the  emphasis  in  arithmetic  is  now  placed  upon 
the  practical  values  of  the  subject.  That  does  not  mean 
that  all  the  older  aims,  as  discipline,  pleasure,  culture, 
and  preparatory  values,  are  now  wholly  ignored ;  but  it 
does  mean  that  the  emphasis  has  shifted  from  these  to 
the  practical  in  the  broad  sense  of  that  term.  But  in  so 
doing,  whatever  claims  of  recognition  any  of  the  older 
values  have  may  be  taken  care  of  when  teaching  the 
subject  from  the  practical  standpoint. 

But,  like  the  term  "formal  discipline,"  the  term  "prac- 
tical values"  is  very  much  misunderstood  and  misused. 
Some  would  view  the  practical  side  as  relating  to  nothing 
but  the  ability  to  do  the  computing  necessary  in  one's 
vocation.  Others  think  that  it  relates  to  fitting  one  for 
some  specialized  vocation,  usually  a  clerkship  in  some 
commercial  pursuit.  And  there  are  still  others  that 
consider  the  practical  as  referring  to  any  real  problem 
met  in  any  of  the  world's  varied  activities.  Many  of 
the  problems  that  have  found  their  way  into  the  schools 
in  recent  years  under  the  guise  of  "practical"  are  just  as 
impractical  and  far  more  ridiculous  than  some  of  the 


THE   AIM  OF   A    COURSE  IN   ARITHMETIC      13 

problems  of   the  past  which  were   given  for  recreation 
and  mental  gymnastics. 

But,  by  a  more  careful  analysis  of  the  practical  values, 
from  the  standpoint  of  the  general  user  of  arithmetic  hi 
society  rather  than  the  needs  of  the  man  in  some  specialized 
vocation,  it  is  easily  seen  that  the  following  things  are 
practical  to  the  average  person  in  any  walk  of  life : 

1.  Efficiency  in  computation. 

2.  A  social  insight  into  business  and  industrial  practices 
that  will  enable  one  to  interpret  references  to  such  practices 
met  in  general  reading  or  in  social  and  business  intercourse. 

3.  Power  to  express  and  to  interpret  the  numerical 
expressions  of  the  quantitative  relations  that  come  within 
our  experiences. 

4.  The  habit  of  seeing  such  relations,  particularly  those 
that  are  vital  to  our  welfare. 

While  to  a  degree  these  four  abilities  are  of  importance 
to  all,  their  values  vary  with  the  different  users  of  the 
subject. 

EFFICIENCY  IN  COMPUTATION 

Computation  is  not  an  end  in  itself,  but  a  means  to  an 
end.  Just  as  ability  to  form  the  letters  of  the  alphabet 
is  necessary  if  we  are  to  express  our  thoughts  in  writing, 
but  useless  unless  we  have  thoughts  to  express,  so  ability 
to  compute  is  of  no  value  unless  we  know  what  process 
to  apply  to  a  problem,  for  there  is  no  relation  between 
ability  to  compute  and  ability  to  reason  out  what  process 
to  apply  to  a  given  problem.  This  is  sometimes  mis- 
understood, and  drill  hi  pure  computation  is  overem- 


14  THE    TEACHING  OF   ARITHMETIC 

phasized  in  the  thought  that  we  are  thus  teaching  arith- 
metic. 

Since  mechanical  ability,  like  computation,  is  more 
easily  measured  than  other  results  of  teaching,  this  phase 
is  likely  to  be  overemphasized,  particularly  in  those 
schools  where  "standard  tests"  are  too  closely  relied  upon 
in  measuring  the  work  of  the  school.  However,  there  is 
need  for  accuracy  and  reasonable  speed  in  using  the  four 
fundamental  processes  in  whole  numbers,  fractions,  and 
decimals ;  skill  in  mental  calculations ;  and  ability  to  see 
approximate  results  without  a  pencil. 

The  absolute  control  of  the  decimal  point  must  be 
developed.  The  lack  of  such  a  control  is  the  cause  of 
numerous  errors.  Recently  I  saw  a  class  attempt  to 
find  the  cost  of  2.9  cubic  feet  of  gas  which  they  had  used 
hi  an  experiment,  knowing  that  the  price  was  $.90  per 
1000  cubic  feet.  The  class  was  divided  into  about  three 
equal  groups  giving  26.1^f,  2.61  ff,  and  .261  £  respectively 
as  the  answer. 

Efficiency  in  computation,  however,  requires  more 
than  a  mere  automatic  control  of  number  facts  and 
processes.  Through  a  knowledge  of  the  fundamental 
principles  involved,  pupils  of  the  upper  grammar  grades 
should  have  the  power  to  see  relations  that  will  save 
figures  and  even  whole  processes.  Thus,  if  one  is  to  find 
what  per  cent  a  gain  of  $2,346,275  is  of  the  total  sales 
of  $86,724,342,  he  should  be  able  to  see  that  not  half  of 
the  figures  given  need  to  be  used  in  the  calculation.  Or, 
if  he  is  to  find  the  interest  at  6  per  cent  by  multiplying 
the  principal  by  the  number  of  .days,  pointing  off  three 


THE   AIM   OF   A    COURSE   IN   ARITHMETIC       15 

more  decimal  places,  and  dividing  by  six  —  a  very 
common  method  —  he  should  see  that,  if  the  number  of 
days  is  a  multiple  of  six,  the  division  may  be  saved  and 
the  result  found  by  multiplying  i>y  one  sixth  of  the  num- 
ber of  days,  and  pointing  off  three  more  places. 

SOCIAL  INSIGHT  INTO  BUSINESS  AND  INDUSTRIAL 
PRACTICES 

The  proper  study  of  arithmetic  should  develop  an  ability 
to  interpret  and  comprehend  the  quantitative  problems 
that  arise  in  the  world's  varied  activities.  Whether  one ' 
is  to  be  actively  engaged  in  a  commercial  or  industrial 
vocation  or  not,  in  his  daily  reading  and  in  intercourse 
with  others  he  will  meet  references  to  many  of  the  prob- 
lems of  commercial  and  industrial  activities,  and  he  must 
be  able  to  interpret  them  if  he  is  to  take  an  important 
place  in  society.  So  the  problems  of  arithmetic  are  now 
taking  on  a  much  more  real  nature  than  those  of  the  past 
did.  In  a  discussion  of  borrowing,  loaning,  and  invest- 
ing money,  of  taxes,  insurance,  etc.,  the  indirect  prob- 
lems of  the  past  are  fast  giving  way  to  those  problems 
that  one  actually  needs  to  ask  himself  in  any  of  these 
phases  of  social  life. 

For  some  students,  the  power  to  interpret  and  compre- 
hend such  problems  has  no  further  practical  value  than 
the  enrichment  of  their  understanding  of  the  life  about 
them.  But  to  a  vast  majority  of  the  students,  there  is 
a  more  direct  vocational  value;  for,  in  our  present  com- 
mercial and  industrial  competition  and  struggle,  other 
things  being  equal,  the  one  having  the  keenest  sense  of 


16  THE   TEACHING  OP  ARITHMETIC 

number  relations   has  by   far  the  greatest   chance  of 
success. 

POWER  TO  EXPRESS  AND  INTERPRET  QUANTITATIVE 
RELATIONS 

The  relations  between  numbers  representing  magnitudes 
are  either  differences  or  ratios.  Of  the  two  means  of 
expressing  relations,  the  ratio,  expressed  as  a  fraction 
or  a  per  cent,  gives  the  most  definite  or  vital  idea  or 
picture.  For  example,  to  say  that  in  1915  the  world's 
production  of  cotton  was  14,126,500  bales,  of  which  the 
United  States  produced  all  but  2,934,700  bales,  creates 
no  mental  picture  and  means  but  little  to  us.  But,  to 
give  a  more  definite  picture,  we  say  that  the  United  States 
produced  about  four  fifths  of  the  world's  crop  of  cotton 
in  1915. 

The  task  of  developing  power  to  see  and  to  interpret 
quantitative  relations  is  a  most  difficult  one.  It  can  only 
be  done  by  making  a  constant  application  to  the  quantita- 
tive aspects  of  life,  ^  While  many  feel  that  we  have  lost 
a  means  of  developing  such  power  by  discarding  the  kind 
of  problems  found  in  the  older  types  of  "mental"  and 
"intellectual"  arithmetic,  it  seems  reasonable  that  a 
closer  correlation  of  arithmetic  with  the  actual  quantitative 
situations  that  arise  in  other  school  activities  and  in  the 
everyday  life  of  the  child  will  come  more  nearly  develop- 
ing an  ability  that  is  of  real  use  in  interpreting  the  quan- 
titative phases  of  life  and  in  doing  the  world's  work  than 
can  possibly  come  through  artificial  problems  made  up 
merely  for  mental  gymnastics. 


THE   AIM  OF   A   COURSE   IN   ARITHMETIC       17 

ELIMINATION  OF  TOPICS 

A  discussion  of  the  values  of  a  course  in  arithmetic 
naturally  leads  to  a  discussion  of  what  is  worth  while  in 
the  course  and  what  may  be  eliminated  as  not  con- 
tributing to  any  of  the  purposes  for  which  arithmetic 
is  taught.  While  there  can  be  no  definite  statement 
made  that  will  meet  every  case,  since  much  depends  upon 
the  individual  user  of  arithmetic,  but  few,  if  any,  in  the 
ordinary  walks  of  life  will  find  much  use  for  the  following : 

1.  The  greatest  common  divisor.  —  The  only  application 
of  the  greatest  common  divisor  that  is  found  in  elementary 
arithmetic  is  its  use  in  reducing  a  fraction  to  its  lowest 
terms.    But,  if  the  greatest  common  divisor  is  to  be  found 
by  the  factoring  method,  it  will  be  more  economical  to  divide 
out  the  common  factors  as  they  are  found  than  to  take 
their  product  and  then  divide  both  terms  by  this  product. 

2.  Addition  and  subtraction  of  fractions  with  large  or 
unusual  denominators.  —  On  account  of  the  nearly  uni- 
versal use  of  decimals,  the  addition  and  subtraction  of 
fractions,  except  the  very  simplest  ones,  rarely  ever  occurs 
in   practical   work.    Even   those   fractions   and   mixed 
numbers  that  are  added  and  subtracted  usually  arise 
from  expressing  some  number  as  5  bu.  3  pk.,  3  Ib.  7  oz., 
etc.,  in  terms  of  a  single  unit  as  5f  bu.,  3^  Ib.,  etc. 

The  fractions '  with  larger  terms,  used  to  express  a 
ratio,  are  never  added.  Such  ratios,  however,  are  usually 
expressed  decimally. 

3.  The  least  common  multiple.  —  In  elementary  arith- 
metic, the  only  application  of  the  least  common  multiple 


18  THE   TEACHING  OF  ARITHMETIC 

of  two  or  more  numbers  is  its  use  in  reducing  fractions 
to  common  denominators.  So,  if  only  the  fractions  found 
in  the  practical  problems  of  everyday  life  are  given,  the 
least  common  multiple  is  not  needed,  for  the  common 
denominators  can  be  found  more  quickly  by  inspection. 

4.  The  more  complex  forms  of  complex  fractions.  —  It 
has  become  common  among  writers  in  recent  years  to 
recommend  unreservedly  the  elimination  of  all  complex 
fractions.  However,  a  careful  examination  into  the 
nature  and  use  of  the  complex  fraction  causes  a  modifica- 
tion of  such  a  recommendation.  The  relations  wanted 
among  fractions  or  mixed  numbers  are  quite  as  common 
as  those  among  whole  numbers;  and  a  complex  fraction 
is  but  a  ratio  or  an  expressed  division  between  two  frac- 
tions or  mixed  numbers  or  a  whole  number  and  a  fraction 
or  a  mixed  number.  Hence,  the  following  ratios,  ex- 
pressed as  fractions,  are  as  apt  to  occur  as  those  between 
whole  numbers,  namely : 

i      5i      ±      1      2i      M 

?       8i'       2*'       3i'         5'        f 

It  may  be  urged  that,  since  these  are  but  special  forms 
of  expressing  a  division,  why  call  them  fractions,  and 
why  not  simplify  them  as  in  ordinary  division  of  fractions. 
But  perhaps  by  writing  them  in  this  form  and  applying 
the  principle  that  Multiplying  both  terms  of  a  fraction  by 
the  same  number  does  not  alter  the  value  of  the  fraction  is 
the  most  economical  way  of  simplifying  such  ratios. 
Thus,  at  sight,  the  above  "complex  fractions"  become: 

&      ii,      &,      f,      TS,      and      ^. 


THE  AIM  OF   A    COURSE  IN   ARITHMETIC     19 

5.  Obsolete  tables  and  those  used  in  specialized  vocations. 
—  Under  these  tables  would  be  included  apothecaries' 
weight,  troy  weight,  surveyors'  measure,  folding  of  paper, 
foreign  money,  and  also  the  gill,   furlong,  rood,  dram, 
quarter  in  avoirdupois  weight,  etc. 

6.  Impractical    reduction    in    denominate    numbers.  — 
Reductions  of  more  than  one  or  two  steps  rarely,  if  ever, 
occur.    One  might  have  occasion  to  reduce  feet  to  inches 
or  inches  to  feet,  but  there  is  not  apt  to  arise  a  need  of 
changing  miles  to  inches  or  inches  to  miles. 

7.  Addition,    subtraction,   multiplication,   and    division 
of  compound  denominate  numbers.  —  In  practical  prob- 
lems, a  measure  is  expressed  as  a  whole  number,  a  mixed 
number,  or  a  decimal  in  one  unit  instead  of  as  a  compound 
denominate  number,  before  it  is  added,  subtracted,  mul- 
tiplied, or  divided.    Hence,  these  topics  need  not  receive 
any  attention  in  the  schoolroom. 

8.  The  present  type  of  inverse  problems  in  fractions  and 
percentage.  —  Most  of  the  inverse  or  indirect  problems 
given  in  fractions  or  percentage  could  not  meet  a  real 
need  of  any  one.    They  are  a  sort  of  "hide-and-go-seek" 
kind  given  for  "exercises  in  analysis."    Thus,  in  the  prob- 
lem :    "If  Mr.  A  sold  a  suit  for  $24,  thereby  making  20 
per  cent  of  the  cost,  find  the  cost,"  the  answer  cannot 
meet  a  real  need  of  Mr.  A,  for  he  must  have  known  the 
cost  before   he   could   have  furnished  the  data  of  the 
problem. 

It  may  occur,  however,  that  Mr.  A  has  a  suit  of  which 
he  knows  the  cost  and  wishes  to  know  a  selling  price  that 
will  iyield  a  certain  per  cent  of  itself.  This  would  involve 


20  THE   TEACHING  OF  ARITHMETIC 

the  indirect  type  of  problem.    Usually,  however,  com- 
petition determines  the  price. 

There  are  perhaps  sufficient  reasons  for  bringing  up 
the  inverse  or  indirect  type  of  problems  in  a  high  school 
course;  but  there  is  much  more  profitable  work  for  the 
grammar  school  pupil,  at  least  before  the  eighth  grade, 
than  even  the  real  applications  of  the  indirect  problem. 

9.  The  various  short  methods  of  finding  interest.  —  The 
one  who  has  to  compute  interest  often,  as  a  clerk  in  a 
bank,  makes  use  of  a  book  of  tables.    One  who  computes 
interest  but  infrequently  is  apt  to  forget  any  of  the 
various  short  methods;  but,  if  he  knows  the  meaning  of 
interest,  he  cannot  forget  the  general  method,  —  that  is, 
that  the  principal  multiplied  by  the  rate,  and  that  product 
multiplied  by  the  time  expressed  as  a  fraction  of  a  year, 
gives  the  interest. 

10.  All  inverse  problems  of  interest.  —  The  inverse  or 
indirect  problems  of  interest  do  not  occur  in  practical 
life,  and  do  not  make  more  clear  the  meaning  of  interest, 
hence  they  can  be  justified  only  "for  analysis."    But 
that  justification  only  is  not  sufficient,  for  there  are  plenty 
of  problems  that  furnish  sufficient  drill  hi  analysis  while 
also  serving  a  more  important  purpose. 

11.  Partial  payments.  —  The  subject  is  not  of  sufficient 
importance  to  justify  it  from  any  of  the  purposes  of 
arithmetic. 

12.  Annual  interest.  —  This  cannot  be  justified  from 
the  standpoint  either  of  practical  uses  or  of  social  insight. 

13.  Undue  emphasis  upon  the  discounting  of  interest- 
bearing  notes.  —  Beyond  knowing  the  meaning  of  Bank 


THE  AIM  OF  A   COURSE  IN  ARITHMETIC     21 

Discount  —  that  is,  that  when  a  bank  collects  the  interest 
in  advance,  this  interest  is  called  Bank  Discount  —  drill 
in  finding  discount  is  not  worth  while. 

14.  True  discount. — As  a  business  custom,  this  is  obso- 
lete.   Yet,  it  is  given  in  many  of  the  textbooks  still  in  use. 

15.  Partnership.  —  This  does  not  lead  to  an  insight 
into  business  practices,  for  it  is  obsolete  except  in  the 
smaller  businesses;  and  the  problems  usually  given  lead 
to  an  erroneous  notion  of  the  division  of  profits  even  in 
a  partnership  business. 

16.  Proportion  as  a  general  method  of  solving  problems. 

—  Other  methods  of  analysis  and  solution  are  not  only 
shorter  but  develop  greater  power  to  see  and  express 
quantitative  relations. 

17.  Foreign  and  domestic  exchange. — Except  the  knowl- 
edge of  how  indebtedness  may  be  canceled  by  check  or 
draft  instead  of  by  an  actual  transfer  of  money,  the  sub- 
ject is  of  no  real  practical  value  to  the  general  student  of 
arithmetic. 

18.  The  measurement  of  uncommon  areas  and  volumes. 

—  The  measurement  of  such  areas  and  volumes  as  those 
of  trapezoids,  frustums,  wedges,  spheres,  etc.,  are  so  un- 
common in  practical   everyday  life  that  the  method  of 
finding  them  is  soon  forgotten. 

19.  Square  root  and  the  Pythagorean  theorem.  —  One  of 
these  topics  is  usually  given  as  an  excuse  for  giving  the 
other.    jWhile  they  have  but  little  practical  value  to  the 
general  user  of  arithmetic,  reference  to  them  is  yet  so 
common  in  general  reading  that  a  mere  acquaintance 
with  what  they  mean  may  justify  a  brief  discussion  of 


22  THE   TEACHING  OF  ARITHMETIC 

them.    It  is  better,  however,  to  delay  these  topics  until 
the  high  school  course. 

20.  The  metric  system.  —  Unless  the  metric  system  is 
going  to  be  needed  in  science,  its  teaching  cannot  be 
justified.  It  is  sometimes  urged  as  a  means  of  bringing 
about  a  general  use  of  the  system,  but  it  is  difficult  to 
show  its  advantages  in  an  appealing  way  to  grammar 
school  pupils.  The  teaching  of  the  subject  should  be 
delayed,  then,  until  it  is  met  in  science. 

THE  ESSENTIAL  WORK  IN  ARITHMETIC 

There  are  those  who  seem  inclined  to  think  that  the 
elimination  of  these  twenty  topics  means  a  greatly  re- 
duced time  to  be  given  to  arithmetic,  but  this  is  not  the 
case.  On  every  hand,  one  hears  from  the  business  world 
the  complaint  that  grammar  school,  high  school,  and 
college  students  know  but  little  arithmetic ;  that  is,  that 
the  arithmetic  of  the  schools  does  not  "function"  in 
actual  practice.  On  every  hand  teachers  are  either 
asking  for  or  suggesting  a  remedy.  No  panacea  has  yet 
been  given.  But  it  seems  that  the  following  suggestions 
may  help  bring  about  better  results : 

1.  Develop  better  habits  of  accuracy.  —  From  the  very 
beginning  of  the  work,  pupils  should  be  given  methods  of 
checking  their  work,  and  all  work  should  be  carefully 
checked ;  also  all  work  handed  in  should  be  100  per  cent 
accurate  in  computation.  The  same  standard  of  speed 
cannot  be  expected  of  all  pupils,  but  all  can  work  at  a 
piece  of  computation  until  they  are  assured  of  its  ac- 
curacy, just  as  an  accountant  does. 


THE  AIM  OF  A   COURSE   IN   ARITHMETIC     23 

2.  Develop  reasonable  speed.  —  There  is  some  danger 
that  too  great  an  effort  is  made  to  get  a  fixed  speed.     In 
fact,  too  conscious  haste  upon  the  part  of  the  pupil  may 
lead  to  injurious  results.    The  purpose  of  speed  is  largely 
to  save  time,  so  we  must  ask  if  the  time  saved  is  worth 
the  effort  to  attain  it. 

3.  Encourage   short   methods.  —  Pupils   should   under- 
stand fundamental  principles  that  will  lead  to  a  saving 
of  figures  and  processes.    This  is  a  slow  and  gradual 
growth  that  comes  about  with  greater  maturity  and  a 
greater  insight  into  fundamental  principles  and  numerical 
relations. 

4.  Emphasize  oral  work.  —  In  life,  one  uses  much  more 
oral  than  written   arithmetic.     Pupils  should  learn  to 
work  without  a  pencil  and  to  give  either  exact  answers  or 
very  close  approximations  to  all  types  of  problems  that 
arise  in  everyday  life. 

5.  Habituate    rather   than    rationalize   those  facts   and 
processes  of  frequent  recurrence.  —  When  a  fact  or  process 
is  used  without  variation  and  arises  very  frequently  in 
the  life  of  the  average  student  of  arithmetic,  drill  should 
continue  until  the  recalling  of  the  fact  or  process  becomes 
a  habit.    But  the  reason  for  the  fact  or  process  may  well 
be  delayed  or  omitted  entirely.    Thus,  the  primary  facts 
and  the  fundamental  processes  with  whole  numbers  must 
be  made  automatic,  but  to  discuss  why  we  "carry"  or 
why  we  "borrow"  is  unnecessary  when  the  work  is  first 
taken  up. 

6.  Avoid   stereotyped   solutions.  —  The   solution    of   a 
problem  should  never  be  an  act  of  memory  but  always 


24  THE   TEACHING  OF  ARITHMETIC 

a  process  of  reasoning.  Otherwise,  the  pupil  is  not 
gaining  power  to  see  relations  but  merely  developing  skill 
in  juggling  figures. 

7.  Use  the  quantitative  situations  that  arise  in  all  other 
activities  in  or  out  of  school.  —  Every  process  as  learned 
should  find  an  immediate  use  in  answering  some  real 
inquiry  of  the  pupil  about  the  quantitative  relations  that 
are  vital  to  his  interests. 


CHAPTER  III 
COUNTING 

ROTE  AND  RATIONAL  COUNTING 

THE  idea  of  number  is  coeval  with  the  human  race. 
With  the  ability  to  distinguish  differentia  —  that  this 
is  not  that  —  originated  the  concept  of  number.  But 
the  first  concept  is  the  "how  many"  idea  of  number, 
the  "how  much."  The  basis  for  the  number  facts  is 
counting.  Hence,  the  first  step  in  the  teaching  of  num- 
bers is  counting,  first  by  rote,  then  rationally.  In  other 
words,  the  child  first  gets  the  order  or  sequence  of  the 
numbers  and  is  able  to  say,  "one,  two,  three,  four,  five, 
six,"  etc.,  without  being  able  to  recognize  any  of  these 
numbers  of  things.  Then  from  objects  he  learns  the 
meaning  of  these  names;  that  is,  the  counting  becomes 
rational. 

When  pupils  have  trouble  in  getting  the  order  of  the 
numbers  fixed,  or  when  they  have  developed  a  wrong 
order,  as  "one,  two,  three,  five,  eight,  nine,"  etc.,  rote 
counting  is  often  taught  or  corrected  through  the  use  of 
little  rhymes,  as : 

One,  two,  three,  four,  five, 

I  caught  a  hare  alive ; 
Six,  seven,  eight,  nine,  ten, 

I  let  hun  go  again. 
25 


26  THE    TEACHING  OF   ARITHMETIC 

And, 

One,  two,  Five,  six, 

Buckle  my  shoe ;          Pick  up  sticks ; 

Three,  four,  Seven,  eight, 

Shut  the  door ;  Lay  them  straight ; 

Nine,  ten, 

I  see  a  fat  hen. 
Or, 

One  little,  two  little,  three  little  Indians, 
Four  little,  five  little,  six  little  Indians, 
Seven  little,  eight  little,  nine  little  Indians, 
Ten  little  Indian  boys. 

When  a  child  can  call  the  names  of  the  numbers  in 
order,  the  next  step  is  to  teach  rational  counting ;  that  is, 
through  counting  objects  about  the  room,  the  child  must 
see  the  meaning  of  the  names  he  has  learned. 

In  having  a  pupil  count  objects,  the  teacher  should 
take  care  that  he  doesn't  get  the  idea  that  the  second  ob- 
ject is  "two,"  the  third  one  "three,"  etc.  This  is  avoided 
by  presenting  groups  and  raising  the  question  of  "how 
many"  and  then  finding  it  by  counting.  Thus,  present- 
ing three  things  as  I  I  I  ask,  "How  many?"  The 
child  counts,  "One,  two,  three,"  and  says,  "There  are 
three  of  them."  Presenting  four  things,  ask,  "How 
many  ?  "  —  "  one,  two,  three,  four,  —  there  are  four  of 
them,"  etc.  Make  use  of  objects  in  the  room.  Ask  how 
many  windows,  how  many  children,  how  many  pictures, 
how  many  books,  etc.,  until  the  children  can  apply  their 


COUNTING 


27 


rote  counting  to  finding  the  "how  many"  of  the  things 
about  them.  Use  objects  in  which  the  children  are  inter- 
ested, not  those  made  for  the  occasion,  as  splints,  beads,  etc. 

THE  NAMES  OF  THE  FIGURES 

Before  teaching  any  of  the  primary  combinations,  the 
pupil  should  recognize  the  names  and  meanings  of  the 
figures.  Thus,  he  must  associate  the  figure  with  the 
name  and  with  the  objects,  as  I  I  I  I  I,  five,  5; 
I  !  I,  six,  6;  etc.  The  teacher  will  devise  many  interesting 
ways  of  presenting  this  so  as  to  fix  these  relations.  One 
very  enjoyable  way  to  fix  them  is  through  a  little  "  match- 
ing game."  To  play  this  game,  take  thirty  cards  —  ten 
of  them  containing  the  figures,  1,  2,  3,  4,  5,  6,  7,  8,  9,  10 
(one  figure  on  each  card) ;  ten  others  containing  the 
names,  one,  two,  three,  etc.;  and  ten  others  containing 
objects  as  dots  arranged  as  on  domino  cards.  These 
cards  are  distributed  among  the  pupils  and,  as  a  pupil 
is  named,  he  runs  to  the  front  of  the  room  and  shows  his 
card.  The  other  pupils  "match"  the  card,  by  coming 
up  and  standing  in  line.  Thus,  if  a  pupil  shows  "six," 
it  is  matched  by  6  and  III,  shown  below. 


In  this  and  other  ways,  a  pupil  is  soon  able  to  associate 
the  figures  with  the  names  and  with  the  things  which  they 
represent. 


28 


THE    TEACHING  OF   ARITHMETIC 


3 

13 

4 

14 

5 

15 

6 

16 

7 

17 

8 

18 

9 

19 

COUNTING  TO  TWENTY  AND  TO  ONE  HUNDRED 

Since  our  number  system  is  based  upon  a  scale  of  ten, 
counting  by  ones  to  ten  is  the  first  unit  of  instruction  in 
counting.  This  is  followed  by  counting  to 
twenty. 

Counting  from  10  to  20  may  be  accom- 
plished at  the  same  time  that  the  written 
forms  are  given.  In  this  way,  the  child 
sees  the  similarity  between  3  and  13,  4  and 
14,  etc.  At  the  same  time,  he  gets  the 
sound  "three,"  "thirteen";  "four,"  "four- 
teen" ;  etc.,  and  sees  that  by  adding  "teen" 
the  order  is  like  the  order  he  already  knows. 

Eleven  and  twelve  must  be  taught  separately,  and 
there  are  no  sound-aids  to  help  the  memory  as  in  "teens." 

The  next  step  is  counting  by  tens  to  one  hundred. 
This  is  easily  accomplished  by  calling  the  child's  attention 
to  the  similarity  in  sound  between  "two" 
and  "twenty";  "three"  and  "thirty"; 
"four"  and  "forty";  "five"  and  "fifty"; 
etc.  Thus  he  sees  that  it  is  very  much  like 
counting  by  ones  to  ten  if  the  suffix  ty  is 
annexed  to  each  of  the  number  names 
from  two  to  nine.  This,  too,  may  be  ac- 
complished by  using  the  written  forms  as 
in  the  "teens." 

The  third  step  is  the  filling  of  each  dec- 
ade with  the  one,  two,  three,  etc.,  as  twenty-one,  twenty- 
two,  twenty-three,  etc. 


2 

20 

3 

30 

4 

40 

5 

50 

6 

60 

7 

70 

8 

80 

9 

90 

COUNTING  29 

READING  AND  WRITING  NUMBERS 

When  a  child  is  first  taught  to  read  and  write  num- 
bers, the  place-value  feature  of  our  system  of  writing 
numbers  should  not  be  emphasized  even  if  it  is  taught 
at  all.  He  should  be  taught  to  read  and  write  numbers 
as  he  reads  and  writes  words.  Thus,  while  writing  36, 
say,  "this  is  thirty-six,"  etc.  The  child  thus  sees  that 
the  3  suggests  the  thirty  and  the  6  the  six,  and  easily  reads 
any  two-figured  number.  In  the  same  way  he  is  taught 
to  read  hundreds,  thousands,  etc.  The  real  significance 
of  "place  value"  should  be  reserved  until  the  child  is 
more  mature  and  until  such  knowledge  is  needed  hi  order 
to  understand  the  why  of  the  written  processes. 


CHAPTER  IV 

THE  PRIMARY  FACTS  OF  ADDITION:    WRITTEN 
ADDITION 

ASIDE  from  a  few  primary  facts  that  have  been  picked 
up  incidentally  during  the  work  in  counting  in  the  first 
year,  the  work  of  teaching  the  primary  facts  seems  to  be 
more  economically  done  by  teaching  but  one  process  at' 
a  time,  rather  than  by  teaching  all  processes  about  a  given 
number  at  once,  as  in  the  Grube  plan.  While  teachers 
differ  as  to  the  best  order  in  which  to  teach  the  primary 
facts  of  addition,  whether  to  teach  the  tables  of  ones, 
twos,  threes,  etc.,  or  to  teach  all  of  the  addition  facts 
about  a  certain  number,  the  order  is  much  less  important 
than  the  nature  of  the  drill  work  that  is  needed  to  make 
the  facts  automatic.  However,  the  order  is  important 
enough  to  merit  serious  thought  and  discussion. 

THE  FIRST  GROUP  OF  PRIMARY  FACTS 

There  are  forty-five  possible  combinations  of  two  one- 
figured  numbers.  These  are  called  "the  forty-five  pri- 
mary facts  of  addition." 

The  first  group  of  these  facts  to  be  thoroughly  fixed, 
whatever  order  is  used,  consists  of  the  twenty-five  com- 
binations whose  sums  do  not  exceed  ten.  This  is  not 

30 


THE   PRIMARY   FACTS  OF   ADDITION          31 

only  the  logical  and  psychological  division  of  the  facts, 
but  an  economic  one  in  the  matter  of  learning  them. 
The  most  economic  order  seems  to  be  to  follow  the  tables 
in  the  ones  and  twos  at  least;  for,  when  a  child  can 
count  rationally  by  ones,  he  really  knows  nine  of  the 
first  twenty-five  facts;  that  is,  he  sees  that  to  add  one 
is  to  call  the  next  number  in  the  order  of  counting.  How- 
ever, some  drill  in  calling  these  sums  when  seeing  the 
written  figures  is  necessary. 

When  any  new  sum  is  learned,  the  pupil  should  see 
it  written.  This  written  form  should  be  placed  before 
him  for  days  until  it  leaves  a  mental  picture.  Thus,  when 
the  ones  are  presented,  the  pupil  should  see : 

12345678         9 

!!!!!!!!_L 

2"34~5678910 

This  should  leave  such  a  mental  image  that  when  seeing 

123456789 

1III1IIII 

he  can  automatically  name,  or  write,  the  sum.  Each 
combination  learned  must  be  recognized  instantly  when 
written  in  either  of  the  two  possible  forms  shown  below : 

31         41         51         61         71 
1     3         1     4          1     5         1     6         1     7 

In  order  to  see  the  meaning  of  addition,  the  pupil 
should  find  the  first  group  of  twenty-five  facts,  through 
counting  objects.  Thus,  in  the  table  of  twos,  2+2, 


32  THE   TEACHING  OF   ARITHMETIC 

3+2,    4+2,    etc.,  to   find   4+2,  he   should   take   four 
objects,  then  picking  up  two  others,  count  /'four,  five, 

4 

six"  and  write    2_.     In  this  way  he  should  discover  for 

6 
himself  the  facts  of  each  table  up  to  5+5. 

THE  USE  OF  OBJECTS 

Objects  are  used  to  fix  the  meaning  of  addition,  not  to 
fix  the  facts  themselves;  that  is,  through  this  objective 
discovery  of  the  facts,  the  pupil  sees  clearly  what  is  meant 
by  "three  and  four  are  seven,"  and  is  able  to  use  these 
facts  in  applications  to  problems  within  the  range  of  his 
experiences.  When  these  facts  have  thus  been  rational- 
ized, the  objects  have  served  their  purpose  and  their  use 
should  be  discarded.  In  the  drills  that  follow,  if  a  pupil 
miscalls  a  fact,  the  right  result  should  be  shown  or  told 
him  at  once.  Nothing  is  gained  by  having  him  find  the 
result  again  through  counting.  In  fact,  to  have  him  do 
so  is  to  establish  a  "counting  habit"  that  must  be  broken 
up  later.  If  6+3  is  given  incorrectly,  6+3  =  9  should 
be  shown  at  once. 

THE  ORDER  AND  NATURE  OF  DRILLS 

The  drills  must  build  up  a  mental  imagery  in  the 
child's  mind.  Hence,  the  first  drills  should  be  sight 
drills;  that  is,  drills  in  which  the  child  sees  the  figures. 
In  these  drills  the  figures  should  be  written  in  column 
form  as  ^iey  are  to  appear  later  in  written  work. 

These  sight  drills  form  a  mental  imagery  so  that  in 


THE   PRIMARY   FACTS   OF   ADDITION          33 

pure  mental  drills,  drills  in  which  the  figures  are  not  seen, 
the  child  calls  results  from  this  jnental  picture.  There 
are  some  children,  however,  who  learn  the  combinations 
more  easily  by  hearing  and  repeating  the  facts  as  "five 
and  three  are  eight";  hence,  this  kind  of  drill  should 
also  find  a  place,  but  not  as  important  a  place  as  the  sight 
drill. 

THE  IMPORTANCE  AND  EXTENT  OF  DRILLS 

The  pupil  must  have  automatic  control  of  all  the 
primary  facts  and  all  the  related  facts  needed  in  the 
fundamental  processes.  It  is  not  sufficient  that  he  knows 
the  meaning  of  "four  and  five"  and  can  find  it  through 
counting,  or  even  recall  it  after  a  moment's  time ;  but  he 
must  know  it  automatically.  Just  as  he  calls  the  words 
in  a  reading  lesson  without  consciously  recalling  the 
letters  of  each  word,  so  he  must  be  able  to  call  sums, 
products,  etc.,  without  being  conscious  of  the  numbers 
represented  by  the  figures. 

The  drill  work  necessary  to  attain  this  need  not  be 
dull  and  repressing  in  the  least.  It  may  be  made  the 
most  enjoyable  part  of  the  work.  An  appeal  to  the  play 
instinct,  the  use  of  scoring  games,  flash  cards,  charts, 
number  downs,  races,  playing  store,  and  various  games 
stimulate  a  keen  interest  in  the  work.  This  subject  is 
discussed  in  Chapter  VIII. 

PREPARING  FOR  THE  SECOND  GROUP  OF  PRIMARY  FACTS 

Before  proceeding  beyond  ten,  all  the  subtraction  facts 
corresponding  to  the  first  group  of  addition  facts  must 


34  THE   TEACHING  OF  ARITHMETIC 

be  made  automatic.    The  methods  and  devices  used  in 
teaching  the  subtraction  facts  are  given  in  Chapter  V. 

The  success  of  the  method  given  below  for  teaching 
the  remaining  twenty  facts  of  addition  requires  automatic 
control  of  the  following  subtraction  facts : 

10     10     98877665544332 
_i_§.!!-?!£!^!21^^2J_ 

This  will  be  evident  from  the  method  shown  below. 

The  next  step  in  the  preparation  is  to  bring  out  the 
meaning  of  "teen."  That  is,  show  the  pupils  that 
thirteen,  fourteen,  fifteen,  etc.,  mean  three  and  ten,  four 
and  ten,  five  and  ten,  etc. 

Let  the  pupils  then  tell  at  sight 

10        10        10        10        10        10        10 
_9       _i       J-      J>       _!      _i      _?. 

Let  10+1  =  11  and  10+2  =  12  follow  this  work,  since 
the  sums  are  not  called  "oneteen"  and  "twoteen,"  but 
eleven  and  twelve. 

DEVELOPMENT  OF  THE  REMAINING  TWENTY  FACTS 

By  the  plan  here  presented,  the  pupil  makes  the  remain- 
ing tables  much  more  easily  than  through  counting,  and 
by  a  method  much  more  useful  to  him  as  a  means  of  re- 
taining the  facts.  In  fact,  to  find  through  counting  that 
eight  and  nine  are  seventeen  in  no  way  aids  the  memory 
in  retaining  this  fact.  The  child  is  unable  to  recognize 
the  number  of  objects  in  such  large  groups,  so  the  finding 
of  the  fact  in  this  way  has  no  advantage  whatever  in  fur- 


THE   PRIMARY   FACTS  OF   ADDITION 


35 


nishing  a  mental  imagery  to  aid  the  memory.  The  purpose 
of  finding  the  first  group  through  counting  was  not  to 
help  fix  the  facts,  but  to  make  clear  the  meaning  of 
addition.  But  addition  should  be  thoroughly  clarified 
by  this  tune  and  the  objects  should  have  served  their 
purpose. 

Begin  the  remaining  twenty  facts  with  the  "nines." 
Write  upon  the  blackboard  the  following : 


Now  take  some  one  tof  these  and,  by  the  use  of  objects, 
show  that  one  less  than  the  number  added  to  nine  is  the 
number  of  "teen"  in  the  sum.  Thus: 


Through  drill,  train  the  pupils  to  form  the  habit  of 
looking  at  the  number  to  be  added  to  9  and  thinking 
"one  less"  as  the  number  of  "teen."  Thus  pointing  to 
the  numbers  written  below,  say  "Add  to  nine." 

57648 

As  the  teacher  points  to  them,  the  pupils  subtract 
one  from  each  and  say :  "fourteen,"  "sixteen,"  "fifteen," 
"thirteen,"  "seventeen."  The  pupil  is  now  ready  to 
drill  upon  the  tables  as  written  above. 

Take  up  the  table  of  "eights"  in  the  same  way  showing 


36  THE   TEACHING  OF  ARITHMETIC 

that  two  less  than  the  number  added  to  8  shows  the  num- 
ber of  "  teen  "  in  the  sum.  t 

Using  charts,  flash  cards,  games,  playing  store,  etc., 
as  in  the  first  group,  drill  until  the  "eights"  and  "nines" 
can  be  given  automatically. 

They  are  the  following  combinations  : 

9998898 

£  1  £  1  2  Z  2 

8989899 


It  will  be  found  that  these  fourteen  facts,  usually 
the  most  difficult  to  teach,  are  as  easily  fixed  as  the 
simple  facts  of  the  first  group. 

There  are  yet  six  facts  to  teach.    They  are  : 

777766 

_!    Ji    JL    _!    JL    JL 

14        13        12        11        12        11 

• 

These  may  be  given  upon  cards  and  the  pupil  asked  to 
find  through  counting  that  they  are  correct.  The  chief 
thing,  however,  is  not  the  finding  that  they  are  correct, 
but  the  fixing  of  them  as  mental  pictures. 

ADDING  ZEROS 

While  to  add  zero  is  to  add  nothing  at  all,  and  while 
the  combinations  with  zero  are  not  among  the  "  forty-five 
primary  facts"  of  addition,  some  practice  in  calling  such 


THE   PRIMARY   FACTS   OF   ADDITION 


37 


combinations  is  necessary  in  order  to  give  an  automatic 
control  of  them.  Such  combinations,  however,  need 
not  receive  as  much  tune  as  the  other  facts. 

A  COMPLETE  CHART  OF  PRIMARY  ADDITION  FACTS 

The  child  needs  drill  upon  all  the  possible  ways  of 
writing  the  figures  including  the  zeros.  Thus,  there  are 
"one  hundred  facts"  instead  of  the  so-called  "forty-five 
facts"  when  zeros  are  used.  In  the  following  table,  the 
easiest  group  is  given  first  and  the  hardest  last.  Do  not 
spend  as  much  tune,  then,  with  the  first  part  of  the  table 
as  with  the  last. 


0 

0 

4 

1 

6 

0 

0 

7 

0 

3 

9 

5 

2 

0 

8 

0 

0 

0 

0 

1 

0 

3 

0 

0 

0 

8 

ft 

1 

0 

5 

0 

0 

°. 

0 

4 

0 

6 

9 

2 

7 

1 

1 

1 

4 

1 

7 

6 

1 

2 

1 

1 

1 

2 

3 

1 

8 

4 

3 

5 

5 

9 

2 

8 

1 

5 

1 

1 

3 

2 

9 

6 

7 

1 

3 

4 

1 

4 

1 

5 

1 

1 

2 

3 

2 

4 

3 

8 

2 

6 

7 

4 

3 

5 

7 

2 

2 

5 

6 

3 

2 

3 

3 

4 

6 

2 

5 

2 

5 

3 

2 

3 

2 

2 

3 

4 

7 

3 

2 

6 

8 

7 

9 

4 

6 

4 

4 

8 

7 

9 

4 

3 

6 

2 

8 

3 

4 

9 

7 

8 

5 

9 

2 

7 

6 

6 

8 

3 

7 

4 

5 

9 

4 

9 

4 

8 

9 

3 

4 

8 

4 

9 

6 

9 

5 

7 

6 

9 

5 

8 

9 

5 

7 

7 

8 

6 

8 

9 

7 

5 

6 

8 

-7 

7 

7 

8 

8 

6 

6 

5 

8 

9 

5 

6 

7 

5 

6 

5 

9 

8 

9 

9 

Such  a  chart  as  this  should  be  made  for  permanent  use 
and  also  a  hundred  flash  cards  containing  the  same  num- 
bers should  be  made.  Daily  drill  from  these  for  a  few 
minutes  should  be  kept  up  long  after  they  are  "learned" 
and  when  other  facts  or  processes  are  being  developed. 


38  THE    TEACHING  OF   ARITHMETIC 

DRILLS  THAT  PREPARE  FOR  WRITTEN  WORK 

Ability  to  give  these  primary  facts  automatically  is 
not  the  only  ability  required  in  adding  a  column.    Thus, 
to  add  the  column  in  the  margin,  if  we  add  down, 
9+6  is  the  only  primary  combination.     The  next    9 
is  15+8,  the  next  is  23+4,  and  the  next  is  27+7.     6 
Thus  it  is  plain  that  to  add  columns  the  pupil  must    8 
be  able  to  add  mentally  (without  seeing  the  figures)     4 
a  two-figured   number   to   a   one-figured  number.    T_ 
While  this  is  closely  related  to  the  primary  facts, 
special  drill  upon  such  combinations  is  needed.    Such 
drills  are  sometimes  called  "adding  by  endings."    Thus 
8  added  to  9  gives  17,  a  number  ending  in  7  ;  so  8  added 
to  any  number  ending  in  9  gives  a  number  in  the  next 
decade  ending  in  7.-   While  there  should  be  some  sight 
work  from  charts  of  such  combinations  as 

7       27       47        37       97        67        17        57 


9        29        19        59        89        39        79        49 
66666666      etc. 

for  all  of  the^rimary  combinations,  a  great  deal  of  pure 
mental  work  should  be  done  in  which  the  teacher  should 
point  to  any  one  of  the  nine  digits 

123456789 

and  say,  "Add  17,"  "Add  26,"  "Add  58,"  "Add  43," 
for  all  numbers  from  10  to  100. 


THE  PRIMARY  FACTS  OF  ADDITION          39 


ADDING  SINGLE  COLUMNS 


Before  pupils  are  required  to  add  two  or  more  columns, 
they  should  have  some  skill  in  adding  single  columns, 
for  adding  numbers  of  two  or  more  places  requires,  in 
general,  carrying  and  also  requires  a  greater  tax  upon 
attention;  that  is,  longer  attention  before  a  break  of 
taking  up  a  new  exercise. 

A  great  deal  of  the  drill  work  should  be  sight  drill  from 
charts,  so  that  the  effort  of  recording  the  figures  in  the 
answers  will  not  detract  from  the  attention  needed  for 
the  addition.  It  is  because  pupils  are  plunged  from  the 
table  of  primary  facts,  not  yet  made  automatic,  directly 
into  written  addition,  that  they  develop,  slow,  inaccurate 
habits ;  and,  instead  of  really  adding,  they  find  the  sums 
through  counting.  When  a  pupil  is  slow  and  inaccurate 
in  written  work  or  has  the  "counting  habit,"  there  has 
not  been  sufficient  drill  in  developing  one  or  more  of  the 
three  fundamental  abilities :  (1)  automatic  control  of 
the  forty-five  primary  facts;  (2)  automatic  control  of 
adding  mentally  a  two-figured  number  to  a  one-figured 
number ;  and  (3)  the  habit  of  reading  a  sum  from  a  single 
column. 

WRITTEN  WORK  IN  ADDITION 

There  is  a  question  among  teachers  as  to  whether  the 
written  processes  should  be  rationalized  or  whether  the 
pupil  should  merely  form  the  correct  habit  of  doing  them. 
Since  the  fundamental  processes  are  always  performed 
in  the  same  way  and  arise  so  often  in  all  future  work, 


40  THE   TEACHING  OF  ARITHMETIC 

rationalization  need  not  be  urged  as  a  help  to  the  memory. 
Also,  the  efficient  use  of  the  processes  in  the  solution  of 
problems  depends  upon  a  rationalization  of  the  meanings 
of  the  processes,  not  upon  the  rationalization  of  modes  of 
procedure  —  carrying,  borrowing,  etc.  —  hence,  it  seems 
that  there  is  little  argument  for  rationalization.  Upon 
this  one  principle  all  will  agree,  viz. :  The  important  and 
essential  thing  in  the  fundamental  processes  is  that  the 
pupils  form  correct  habits  in  the  modes  of  procedure; 
that  is,  habituation,  and  not  rationalization,  is  of  chief 
importance. 

There  are  pupils,  however,  to  whom  the  work  is  made 
more  interesting  through  rationalization  and  there  are 
those  who  always  want  some  authority  for  doing  what 
they  do.  Rationalization  need  not  take  much  time. 
Even  if  all  of  the  class  do  not  get  clearly  the  "why," 
the  objective  presentation,  or  some  method  of  rationaliza- 
tion of  a  process,  satisfies  the  pupils  that  there  is  a  reason- 
able basis  for  proceeding  with  the  work  as  we  do,  and  to 
some  it  makes  the  work  more  interesting. 

METHOD  OF  DEVELOPMENT 

Before  the  process  of  addition  can  be  rationalized, 
the  decimal-place-value  feature  of  our  notation  must  be 
understood.  This  can  be  shown  by  splints,  toothpicks, 
or  something  of  this  nature.  First  write  upon  the  board 
two  or  three  two-place  numbers  as  24,  35,  46.  Have  the 
pupils  count  as  many  splints  by  ones.  Then  count  ten 
and  put  a  rubber  band  around  them,  then  another  ten 
in  the  same  way,  and  so  on  until  less  than  ten  are  left. 


THE   PRIMARY   FACTS  OF   ADDITION  41 

If  24  is  the  number  counted,  the  pupil  finds  2  tens  and 
4  ones  left.  If  35,  then  he  finds  3  tens  and  5  ones.  Thus 
he  sees  that  the  2  of  24  stands  for  tens  and  the  4  for  ones, 
etc.  Now,  to  test  the  understanding  of  the  pupils  as  to 
what  you  have  presented,  have  a  pupil  hold  up  any  num- 
ber (up  to  ten)  of  tens,  and  a  number  of  ones  and  have 
the  rest  of  the  class  write  down  the  number  shown.  Thus, 
if  three  tens  and  six  are  shown,  the  pupils  write  36.  In 
this  way  it  is  easy  to  show  the  decimal-place-value  prin- 
ciple of  our  notation. 

Now  there  are  two  principles  involved  in  adding  num- 
bers of  two  or  more  places.  First,  only  like  numbers  can 
be  added.  Hence,  the  numbers  are  so  written  that  the 
ones  will  all  be  in  one  column,  the  tens  in  another,  etc., 
and  each  column  is  added  separately  because  it  is  made 
up  of  like  things.  Second,  when  the  sum  of  any  column 
is  greater  than  ten,  it  is  reduced  to  units  of  a  higher  order, 
and  those  of  the  higher  order  added  to  the  next  column. 
Hence,  the  first  problems  should  include  but  the  first  of 
these  two  principles  in  order  to  introduce  but  one  diffi- 
culty at  a  time.  However,  it  will  be  necessary  to  take 
but  a  few  problems  in  which  there  is  no  carrying,  for  it 
does  not  take  long  to  form  the  habit  of  adding  a  column 
at  a  time  and  putting  the  result  under  the  column  added. 
The  more  difficult  habit  to  fix  is  that  of  working  from  right 
to  left,  for  in  all  other  work  the  child  works  from  left  to 
right. 

Then,  to  rationalize  the  process  of  addition,  suppose 
you  have  written  the  problem  given  in  the  margin.  With 
splints  or  toothpicks  bound  up  in  bundles  of  ten  each 


42  THE   TEACHING  OF   ARITHMETIC 

and  with  loose  ones,  have  a  pupil  hold  before  the  class 

2  tens  and  6  ones,  and  another  pupil,  4  tens  and 

3  ones,  to  represent  the  26  and  43.    Then  let  the    26 
class  tell  how  many  both  pupils  have,  and  thus  see    43 
that  the  69  stands  for  the  6  tens  and  9  ones  held    69 
by  the  two  pupils.    This  need  not  be  presented 
objectively  more  than  two  or  three  tunes.    The  presenta- 
tion should  then  be  followed  with  a  few  days'  drill  in 
adding  two  or  three  numbers  of  two  or  three  places  in 
which  there  is  no  carrying.     All  drill  should  be  under 
the  direct  supervision  of  the  teacher  so  as  to  insure  the 
forming  of  proper  habits.    As  soon  as  pupils  automatically 
work  from  right  to  left  —  that  is,  add  ones  first,  then 
tens,  etc.  —  they  are  able  to  take  up  the  next  class  of 
exercises. 

This  is  done  by  taking  up  some  such  problem  as  the 
one  in  the  margin  and  proceeding  as  before.     Here 
the  class  sees  that  there  are  15  ones.     Have  them    38 1 
separate  the  15  ones  into  1  ten  and  5  ones.     Have    47 
the  class  see,  then,  that  instead  of  7  tens  and  15    85 
ones,  we  have  8  tens  and  5  ones.     But  few  objec- 
tive illustrations  are  needed  to  rationalize  the  carrying 
process.     Drill  should  then  follow  until  the  process  is  fixed. 
It  is  a  waste  of  time  to  present  the  work  objectively  ex- 
cept when  it  is  first  taken  up.     Do  not  require  pupils  to 
"explain"  their  problems  daily  by  the  use  of  splints. 
They  have  seen  that  there  is  a  rational  basis  for  carrying, 
and  that  is  sufficient.     In  fact,  even  this  much  of  the 
process  of  rationalization  is  unnecessary  to  the  efficient 
use  of  the  subject. 


THE   PRIMARY   FACTS  OF   ADDITION  43 

CHECKING  THE  WORK 

The  best  time  to  develop  the  habit  of  checking  the 
computations  is  at  the  time  the  processes  are  first  taught. 
The  pupils  should  feel  that  when  a  process  is  performed 
but  once  the  work  is  but  half  done.  By  reviewing  it  in 
some  way  he  must  become  convinced  of  its  accuracy. 

In  addition,  it  is  best  to  form  the  habit  of  first  adding 
in  a  given  direction  and  checking  by  adding  in  the  opposite 
direction.     This    brings    up    different    combina- 
tions.    Thus,  in  the  exercise  in  the  margin,  by      78 
adding  down,  the  combinations  are  15,  23,  27;      47 
and  9,  13,  22,  25.     By  adding  up,  they  are  12,  19,      98 
27 ;  and  5,  14,  18,  25.     Until  the  pupil  is  pretty    _34 
sure  of  the  result,  the  first  result  should  be  kept    257 
on  a  bit  of   "scratch  paper"  until  it  has  been 
verified  by  the  check. 

DRILLS  IN  WRITTEN  WORK 

Skill  comes  through  much  practice.  Besides  drill  in 
adding  numbers  of  two  or  more  places,  sight  and  mental 
drill  in  the  primary  facts,  adding  by  endings,  and  add- 
ing single  columns  should  be  continued.  But  little  of 
the  drill  work  should  come  from  problems.  There  is  a 
means  of  securing  greater  interest  than  that  gained  through 
problems,  however  real  they  are.  It  is  the  interest  that 
comes  to  the  pupil  from  the  knowledge  that  he  is  develop- 
ing greater  skill  daily.  In  order  to  show  this  clearly, 
exercises  of  the  same  weight  should  be  used  as  a  test 
very  frequently ;  that  is,  exercises  in  adding  four  or  five 


44  THE   TEACHING  OF   ARITHMETIC 

numbers  of  two  or  three  figures  each  should  be  given  two 
or  three  times  a  week,  throughout  the  term,  and  a  record 
kept  of  the  average  number  tried  by  the  class  and  the 
average  number  that  are  correct.  Graphs  of  these  results 
kept  in  sight  will  prove  an  inspiration  to  the  class,  for,  if 
the  drilling  is  done  properly,  there  will  be  a  marked  im- 
provement shown  during  a  month  or  a  term.  Individuals 
should  also  keep  their  own  records  in  the  same  way. 

Not  only  will  pupils  be  interested  to  see  the  graph 
of  then*  progress  gradually  rise,  but  they  will  also  be 
interested  in  watching  the  graph  of  accuracy  approach 
the  graph  of  the  number  of  exercises  tried. 

It  will  be  found  that  the  incentive  from  trying  to  beat 
one's  former  record  and  a  desire  to  see  the  "graph  of 
progress"  rise  will  have  a  greater  appeal  to  the  pupils 
of  the  fourth  grade  and  above  than  to  the  pupils  of  the 
lower  grades. 

Such  a  record  as  that  described  above  will  also  serve 
another  very  important  purpose.  It  will  enable  the 
teachers  to  standardize  their  work.  Through  the  average 
of  a  number  of  such  records,  one  will  have  a  standard 
by  which  he  may  know  what  a  child  of  a  certain  grade 
should  do.  In  other  words,  when  a  child  can  perform  but 
a  certain  number  of  exercises  in  a  certain  time  with  a 
certain  per  cent  of  accuracy,  it  is  clear  that  in  the  par- 
ticular subject  he  belongs  in  the  third,  fourth,  fifth,  etc., 
grade. 

Another  device  to  secure  interest  in  classroom  drill 
in  written  work  is  to  have  the  pupils  call  out  the  order 
in  which  they  finish  their  work.  Thus,  the  first  pupil 


THE   PRIMARY   FACTS  OF   ADDITION          45 

to  finish  an  exercise  calls  "one"  and  writes  "1"  above 
his  work;  the  next  one  calls  "two"  and  writes  "2"  above 
his  work ;  and  so  on.  When  all  are  done  >and  the  answers 
given,  all  who  have  wrong  answers  draw  a  line  through 
their  numbers.  At  the  end  of  the  period,  if  one  has  a 
record  like  1,  1,  3,  4,  it  shows  both  speed  and  accuracy. 
If  one  has  1$,  20,  10,  Z0,  etc.,  it  shows  the  teacher  at  a 
glance  that  the  pupil  is  both  slow  and  inaccurate.  The 
bright  pupils  like  such  a  contest,  and  the  dull  ones  work 
hard  when  they  know  such  a  test  is  to  be  given  once  or 
twice  a  week. 

The  same  device  may  be  used  in  any  kind  of  written 
work  throughout  all  grades. 

THE  NATURE  AND  USE  OF  PROBLEMS 

The  problems  of  the  primary  grades  are  given  chiefly 
for  one  of  two  purposes.  They  are  given  either  to  clarify 
the  processes  or  to  furnish  a  motive  for,  or  interest  in, 
the  work.  Hence,  they  must  be  very  concrete  and  real 
to  the  pupil  and  must  be  those  to  which  he  might  wish 
to  know  the  answer.  While  enough  problems  should  be 
used  to  show  a  need  of  arithmetic,  when  the  purpose  of 
the  instruction  is  to  develop  skill  in  computation,  pure 
abstract  drills  should  be  used.  It  is  much  easier,  too, 
to  make  well-graded  drills  that  will  emphasize  just  the 
facts  or  processes  needed  than  to  get  the  same  drills 
through  problems,  for  a  problem  should  present  real 
conditions.  Hence,  the  size  of  the  numbers  is  regulated 
by  the  nature  of  the  problems,  and  such  drills  are  not  in 
general  as  well  graded  as  the  pure  abstract  exercises. 


CHAPTER  V 

THE   PRIMARY  FACTS  AND   PROCESSES   OF 
SUBTRACTION 

THE  primary  facts  of  subtraction  come  directly  from 
those  of  addition.  Thus,  when  a  child  can  tell  you  that 
3  and  5  are  8,  the  question,  "  3  and  how  many  are  S  ? " 
comes  not  as  a  new  fact,  but  from  the  addition  fact  already 
known.  The  first  drills  should  be  answers  to  such  ques- 
tions as  the  one  above.  In  sight  drill  they  are  written 
as  follows : 

*34**46         *         *       7 

6       *       *     ,5       6       *       *        8        6       *      etc. 

978'787710109 

The  pupil  gives  the  missing  number  that,  with  the 
given  one,  makes  the  sum  below. 

By  subtraction  we  find  the  answer  to  three  types  of 
questions  illustrated  by : 

(a)  5  apples  taken  from  8  apples  leave  how  many 
apples  ? 

(6)  How  many  apples  added  to  5  apples  make  8 
apples? 

(c)  If  one  plate  contains  8  apples  and  another  5  apples, 
how  many  more  in  the  first  plate  ? 

46 


THE   PRIMARY  FACTS  OF  SUBTRACTION       47 

That  is,  Subtraction  is  the  process  of  taking  one  number 
from  another  to  find  how  many  remain;  of  finding  what 
number  must  be  added  to  a  given  number  to  make  a 
given  sum;  and  of  finding  the  difference  between  two 
numbers. 

While  all  these  meanings  should  be  shown  objectively, 
the  pupil  must  see  clearly  that  the  primary  facts  come 
from  addition. 

The  nature  of  the  drill  work  upon  the  primary  facts 
depends  upon  the  method  to  be  used  in  written  work. 
If  the  "addition  method"  is  to  be  used,  the  thought  in 
the  drills  for  14-9  =  5  will  be  "9  and  5  are  14";  while, 
if  the  "taking-away  method"  is  to  be  used,  the  thought 
will  be  "9  from  14  leaves  5." 

THE  ADDITION  METHOD 

In  deciding  upon  a  method  to  use,  the  questions  asked 
are:  "By  which  method  is  there  the  least  liability  of 
error?"  "Which  method  is  more  easily  presented?" 
and,  "By  which  method  can  the  work  be  done  most 
rapidly  ? ' '  There  are  advocates  of  the  "  addition  method ' ' 
who  would  answer,  "The  addition  method,"  to  each  of 
the  above  questions.  However,  there  do  not  seem  to  be 
available  data  to  warrant  such  a  reply.  But  that  there 
are  strong  points  in  favor  of  the  method,  one  must  recog- 
nize. Thus,  it  follows  so  closely  the  work  of  addition 
that  much  of  the  skill  in  addition  is  carried  over  to  sub- 
traction. It  is  also  more  nearly  the  method  used  by 
clerks  in  counting  out  change. 

The  way  in  which  to  present  the  addition  method  is  first, 


48  THE   TEACHING  OF  ARITHMETIC 

to  take  a  problem  in  which  each  figure  of  the  subtrahend 
represents   a  number  less   than   the  corresponding 
number    in    the   minuend.    Thus,    to    present   the    48- 
problem  in  the  margin,  the  pupil  must  have  ac-    23 
quired  the  idea  from  the  drills  upon  the  primary    25 

986 
facts  as,  4  3  2  etc.,  that  the  top  number  is  the  sum  of 

554 

the  other  two.  Like  "a  corn  on  his  chin"  of  Riley's  Man 
in  the  Moon,  which  is  "  a  dimple  turned  over,  you  know," 
so  subtraction  is  "addition  turned  over,  you  know." 
Then  the  thoughts  in  the  above  problem  are :  3  and  5 
are  8,  write  5 ;  2  and  2  are  4,  write  2.  Since  the  tables 
should  have  been  made  practically  automatic  before 
taking  up  the  written  work,  there  is  practically  nothing 
new  to  teach,  for  the  pupil  has  the  habit  of  beginning  at 
the  right-hand  column  and  of  considering  one  column  at 
a  time,  which  was  formed  in  his  work  in  addition.  The 
"why"  of  this  need  not  be  shown.  However,  if  addition 
was  rationalized,  the  reason  for  this  follows;  but  the 
work  may  be  presented  objectively  if  it  is  thought  that 
such  a  presentation  will  answer  the  curious  or  make  the 
work  more  interesting.  Remember  that  the  important 
thing  is  to  establish  proper  habits  of  procedure  and 
to  develop  skill  in  doing  the  work  accurately  and 
rapidly. 

To  develop  the  second  problem,  the  one  in  which  the 
minuend  has  digits  less  than  the  corresponding  digits 
of  the  subtrahend,  review  those  primary  facts  of  sub- 
traction whose  minuends  are  two-figured  numbers,  as : 


Call  attention  to  the  fact  that  in  these,  the  number 
below  is  less  than  the  one  directly  above. 

Then  present  some  such  example  as  the  one  in  the  mar- 
gin.    Call  attention  to  the  4  being  less  than  6,  so  it  must 
be  the  4  of  14,  which  is  the  sum  of  6  and  some  number, 
as  in  the  table  above.    So  ask  "6  and  what  make    54 
14?"      Write   the   answer,  8,  below  the  6.      Now,    26 
since  54  is  the  sum  of  26  and  some  number,  the  4  is    28 
the  4  from  14,  the  sum  of  6  and  8 ;  hence,  the  1  (ten) 
of  14  was  carried  to  2  of  the  given  addend.    So  the  next 
question  is  "  3  and  what  make  5  ? " 

There  are  two  things,  then,  for  the  pupil  to 
observe : 

(1)  When  the  number  above  is  less  than  the  one  below 
it,  picture  it  as  the  right-hand  figure  of  some  number  of 
"teen." 

(2)  In  such  cases,  one  is  always  added  to  the  next  higher 
digit  of  the  addend  (subtrahend). 

In  the  second  example  in  the  margin  we  think  8  521 
and  3  are  11;  write  3.  1  (carried  from  11)  and  .6  268 
and  5  are  12;  write  5.  1  (carried  from  12)  and  2  253 
and  2  are  5 ;  write  2. 

If  "why"  arises,  show  that  it  is  merely  the  carrying 
that  was  done  in  addition,  for  the  top  number  is  the  sum 
of  the  given  number  and  the  number  found. 

In  presenting  this,  or  any  other  process,  the  process 
should  first  be  presented  very  carefully  by  the  teacher 


50  THE    TEACHING  OF   ARITHMETIC 

and  enough  examples  taken  to  show  fully  the  "how." 
Then  the  pupils  should  be  sent  to  the  blackboard  where 
the  first  work  can  be  done  directly  under  the  eye  of  the 
teacher  in  order  that  she  may  detect  any  mistake  in  using 
the  process  before  that  mistake  becomes  a  habit.  If 
seat  work  or  home  work  is  not  given  until  every  pupil 
knows  how  to  perform  the  process,  then  the  only  errors 
made  will  be  the  accidental  errors.  The  purpose  of  further 
drill,  seat  work,  home  work,  etc.,  is  to  develop  greater 
skill  in  handling  the  problems. 

THE  TAKING-AWAY  METHOD 

The  method  in  most  common  use  in  the  schools  in  this 
country  is  the  "  taking-away "  method  of  subtraction. 
There  is  no  trouble  in  presenting  the  first  class  of 
problems ;  namely,  those  in  which  each  number  in  the  48 
subtrahend  is  less  than  the  corresponding  number  in  16 
the  minuend.  Thus,  in  the  problem  in  the  margin,  32 
the  pupil  thinks  6  from  8  leaves  2 ;  write  2.  1  from 
4  leaves  3 ;  write  3.  "'Since  the  pupil  knows  the  tables 
automatically  before  taking  up  written  work,  this  prob- 
lem presents  nothing  new,  for  the  habit  of  subtracting 
one  column  at  a  time  and  of  working  from  right  to  left 
follows  the  habit  established  in  addition  and  is  not  really 
a  new  habit  here.  After  a  few  problems  to  see  that  these 
habits  are  fixed,  further  drill  need  not  be  ,given  before 
taking  up  the  next  class  of  problems;  that  is,  those' 
in  which  the  so-called  "borrowing"  is  necessary.  The 
pupil  is  very  apt  to  ask  why  here,  so  the  teacher  may 


THE  PRIMARY  FACJS   OF  SUBTRACTION       51 

well  present  one  or  two  examples'  objectively.  Thus,  to 
present  the  problem  in  the  margin  objectively,  show 
62  as  6  bundles  of  tens  and  2  ones,  using  toothpicks  62 
or  splints.  Now  ask  a  pupil  to  take  away  7  ones.  27 
He  sees  that,  before  he  can  do  this,  he  must  have  35 
more  than  2  ones.  Therefore,  have  him  take  one 
bundle  of  the  6  tens;  take  off  the  band  and  put  the  10 
splints  with  the  ones,  thus  giving  12  ones.  Now  taking 
away  7  leaves  5;  write  5.  Now  ask  him  to  take  away 
2  tens  from  the  remaining  5  tens,  leaving  3;  write  3. 
After  a  few  objective  presentations  and  a  few  exercises 
without  objects,  the  pupils  should  be  sent  to  the  black- 
board and  should  work  under  the  direct  eye  of  the  teacher 
until  they  can  work  with  small  numbers  with  but  slight 
danger  of  error.  Increase  the  difficulty  of  the  exercises 
very  gradually.  Avoid  exercises  in  which  it  is  necessary 
to  take  one  from  the  next  higher  order  when  the  figure 
of  that  order  is  zero,  as  in  502  —  136,  until  the  pupils 
handle  the  other  simpler  exercises  well.  Next  take  up 
such  an  exercise  having  zeros  in  the  minuend,  as  a  special 
lesson.  Thus,  in  the  problem  above,  show  that  one  is 
taken  from  5,  making  10  tens;  and  that  one  of  the  tens 
is  then  taken,  leaving  9.  This  class  of  exercise  presents 
special  difficulty,  as  may  readily  be  seen,  and  requires 
special  drill.  This  difficulty,  however,  does  not  occur 
in  the  use  of  the  addition  method. 


CHAPTER  VI 


THE  TABLES  DEVELOPED  FROM  ADDITION 

MULTIPLYING  by  a  whole  number  is  only  a  short  way 
of  finding  the  result  of  adding  a  number  of  equal  num- 
bers. Thus,  7X345  is  only  a  short  way  of  writing  and 
finding  345+345+345+345+345+345+345.  Hence,  it 
would  seem  that  the  most  logical  and  psychological  method 
of  developing  the  primary  facts  is  through  addition,  and 
experience  shows  this  to  be  the  most  economic  and  effi- 
cient method. 

Begin  by  selecting  all  the  doubles  from  the  forty-five 
facts  of  addition.  Show  the  class  the  following  drill 
cards  used  in  addition  : 

123456789 


The  results,  of  course,  are  known.  Now  remove  the 
cards  and  ask  the  class  to  tell  you  how  many  of  each 
number  they  saw  on  a  card  and  to  tell  you  the  result. 
The  reply  will  naturally  be,  "two  ones  are  two,"  "two 
twos  are  four,"  "two  threes  are  six,"  "two  fours  are 

52 


53 

eight,"  etc.  Hence,  the  only  new  thing  in  the  "two 
times"  table  is  the  notation.  That  is,  2X1  =2,  2X2=4, 
2X3  =  6,  etc.,  which  should  be  read,  "two  ones  are  two," 
and  not  "two  times  one  equals  two.")  Always  use  lan- 
guage that  gives  the  clearest  picture  of  the  real  meaning 
of  the  process. 

Before  taking  up  multiplication,  the  pupils  should 
have  developed  some  skill  in  column  addition.  They 
should  at  least  be  able  to  add  single  columns  of  four  or 
five  numbers.  Then  to  develop  the  "three  times  "  table, 
meaning  three  of  each  number  have  been  added,  have  the 
class  write  down  three  of  each  of  the  nine  digits  as  follows 
and  find  the  sum.  Thus : 


1 

2 

3 

4 

5 

6 

7 

8 

9 

1 

2 

3 

4 

5 

6 

7 

8 

9 

1 

2 

3 

4 

5 

6 

7 

8 

_9 

3 

6 

9 

12 

15 

18 

21 

24 

27 

When  the  pupil  sees  what  a  time-saver  this  new  process 
of  multiplication  is,  he  will  have  a  motive  for  learning 
the  tables  and  how  to  use  them.     Thus,  to  find 
the  sum  in  the  margin,  let  him  see  that  by  remember-     468 
ing  that  three  8's  are  24,  he  does  not  need  to  add.     4Q8 
Also,  show  him  that  the  second  method  in  the     46g 
margin  is  a  shorter  way  to  write  the  same  thing, 
and  that  he  now  thinks  three  8's,  three  6's,  and     468 
three  4's,  instead  of  seeing  them,  as  in  addition.     JX3 
Thus,  by  this  method  of  development  the  written 
work  can  be  taken  up  with  each  table,  thereby  forming 


54  THE   TEACHING  OF   ARITHMETIC 

a  strong  motive  for  learning  this  new  process  and  also 
furnishing  much  needed  drill  in  fixing  the  facts. 

After  each  table  is  developed  by  addition,  it  is  written 
down  as  multiplication,  thus : 

3X1=3  3X4  =  12  3X7  =  21 

3X2  =  6  3X5  =  15  3X8  =  24 

3X3  =  9  3X6  =  18  3x9  =  27 

These  are  read  "three  1's,"  "three  2's,"  "three  3's,"  etc., 
for  this  is  just  what  the  pupil  sees  in  the  development. 
In  many  textbooks  the  tables  are  written : 

1X3    3X3    5X3    7X3    9X3 
2X3    4X3    6X3    8X3    10X3 

This  is  because  the  tables  were  developed  through  count- 
ing by  3  and  thus  the  notation  accurately  describes  the 
thing  the  pupil  saw  in  the  development.  This  is  the 
table  of  "threes"  and  not  the  "three  tunes  "  table,  which 
was  developed  through  addition. 

After  a  few  tables  have  been  found,  the  word  "times" 
may  be  used,  for  the  expression  will  now  have  a  meaning 
when  the  pupil  knows  that  3X5  means  that  three  5's 
have  been  added. 

This  method  of  development  is  to  make  clear  the 
meaning  of  multiplication,  that  it  is  a  short  process  to 
save  adding  equal  addends.  Hence,  it  is  not  important 
that  all  of  the  tables  be  found  by  the  pupil.  Perhaps 
the  finding  of  the  tables  through  the  "five  times"  table 
is  sufficient  to  fix  the  full  meaning  and  use  of  multipli- 


THE   PRIMARY   FACTS  OF   MULTIPLICATION     55 

cation.  If  the  child's  interest  seems  to  lag  in  the 
making  of  the  tables  after  the  first  four  or  five  •  •  • 
tables  have  been  found,  the  "nine  times"  table  •  •  • 
may  well  be  taken  up  next  as  follows  :  The  pupil  •  •  • 
should  have  been  shown  objectively  or  otherwise  •  •  • 
from  the  first  that  2X3  =  3X2,  3X5  =  5X3,  etc.  •  •  • 
Thus  in  the  diagram  in  the  margin,  in  rows  there  are 
5X3,  while  in  columns  there  are'  3X5. 

Now,  since  2X9  =  18,  9X2  =  18;  since  3X9=27,  9X 
3  =  27;  since  4  X 9  =  36,  9  X  4 = 36.  These  are  known  from 
the  tables  already  learned.  Now,  point  out  that  in  these 
three  products  in  the  "nine  times"  table,  the  tens'  digit 
of  each  product  is  just  one  less  than  the  number  multi- 
plied by  9,  and  that  in  every  case  the  sum  of  the  digits 
in  the  product  is  9.  Now  tell  the  class  that  this  is  always 
so,  and  hence  to  find  9  times  any  number,  as  9X6,  it  is 
merely  necessary  to  write  one  less  than  6  for  the  tens' 
digit,  that  is,  5,  and  for  the  ones'  digit  write  a  number  that 
with  5  makes  9.  That  is,  9  X  6  =  54,  9  X  8  =  72,  9  X  5 = 45, 
etc. 

Now,  if  the  3  times,  4  times,  5  times,  and  9  times  tables 
have  been  developed,  but  six  new  facts  remain.  They 
are :  6x6,  6X7,  6X8,  7x7,  7x8,  and  8X8.  It  is  better 
to  give  these  facts  to  the  child  than  to  have  him  find  them 
through  addition;  or  the  facts  may  be  given  and  the 
pupils  may  prove  them  by  addition.  This  will  be  a  test 
as  to  whether  or  not  they  really  understand  the  meaning 
of  multiplication. 

Drill  must  be  continued  on  the  tables  until  all  the  facts 
can  be  recalled  automatically. 


56  THE   TEACHING  OF  ARITHMETIC 

A  SECOND  METHOD  OF  DEVELOPMENT 

Many  of  the  older  textbooks  and  some  of  the  newer 
ones  develop  the  tables  through  counting  by  2's,  3's, 
4's,  5's,  etc.  The  plan  is  to  learn  to  count  by  2's,  3's, 
etc.,  then  build  up  the  tables  as  follows : 


2 

2 
2 

2 
2 
2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

4 

6 

8 

10 

12 

14 

16 

18 

While,  of  course,  counting  by  2's  is  adding  2's,  yet  to  the 
child  this  does  not  seem  to  be  addition  as  in  the  former 
method,  and  does  not  leave  as  clear  a  meaning  in  the 
child's  mind  of  what  multiplication  really  is.  More- 
over, the  learning  of  a  single  table  does  not  fit  the  pupil 
at  once  to  take  up  written  work  as  is  the  case  in  the 
former  method.  Thus,  the  pupil  could  not  use  the 
above  table  to  find  2x768.  Or,  if  the  3's  were  learned 
in  that  way,  he  could  not  find  3  times  any  number,  as 
3X645,  from  this  table.  There  is  a  further  danger,  too, 
that  pupils  may  rely  upon  counting  to  find  the  products 
in  future  work  rather  than  fix  the  facts  through 
memory. 


THE   PRIMARY   FACTS   OF    MULTIPLICATION     57 


Observe,  also,  that  to  use  the  method  given  here,  the 
tables  are  not  written  in  the  same  order  as  when  de- 
veloped by  the  first  method.  They  would  be  written: 
1X2,  2X2,  3X2,  4X2,  5X2,  6X2,  etc.  The  difference 
in  the  two  methods  of  development  accounts  for  the  two 
ways  of  writing  the  tables. 

This  method  of  development,  then,  is  not  recommended. 

DRILLS  THAT  PREPARE  FOR  WRITTEN  WORK 

Just  as  in  written  addition  it  was  seen  that  the  mere 
learning  of  the  tables  did  not  fit  the  pupil  for  successful 
written  work,  so  the  tables  alone  are  not  sufficient  for 
successful  written  work  in  multiplication.  Thus,  to 
find  3X485,  the  first  product  is  but  3X5,  but  the  next 
one  is  3X8+1,  and  the  next  3X4+2. 

So  there  should  be  drill  upon  announcing  any  product 
plus  any  one-figured  number.  A  simple  device  for  such 
drill  is  a  chart  made  as  follows : 

To  use  the  chart,  merely  point 
to  any  number  whatever  in  each 
column,  going  from  left  to  right. 
Then  let  the  pupil  give  the  product 
of  the  first  two  plus  the  third.  It 
will  be  seen  that  no  possible  com- 
bination in  any  written  work  can 
occur  that  cannot  be  drilled  upon 
from  such  a  chart.  A  five-minute 
drill  daily  upon  such  a  chart  will  aid 
very  greatly  in  efficient  written  work. 


1 

9 

1 

2 

•^ 

8 

2 

3 

* 

7 

3 

4 

6 

4 

5 

X 

5+' 

5 

6 

4 

6 

7 

3 

7 

8 

2 

8 

9 

1 

9 

58  THE    TEACHING  OF   ARITHMETIC 


WRITTEN  MULTIPLICATION 

It  was  shown  above  that  written  multiplication  by 
multipliers  of  one  figure  should  be  taken  up  when  each 
table  is  developed  in  order  to  furnish  a  motive  for 
learning  the  tables  and  to  give  further  drill  that 
is  needed  to  fix  them.  When  a  pupil  sees 
that  instead  of  a  long  problem  in  addition, 

as  in  the  left-hand  margin,  the  same  result     

475  ,     ,       ,  ,          u. %     ..  .     ,,       3325 

may  be  found  by  multiplication,  as  m  the 

right-hand  margin,  he  not  only  understands  what 
multiplication   means,   but    sees   a    very   strong 


3325  »     , 

motive  tor  learning  it. 

In  presenting  written  multiplication,  it  is  necessary 
merely  to  point  out  that,  instead  of  seeing  the  number  of 
equal  addends,  we  think  them;  that  is,  instead  of  seeing 
seven  5's  as  in  addition,  we  think  seven  5's  are  35;  in- 
stead of  seeing  seven  7's,  we  think  seven  7's  are  49,  and 
3  to  carry  are  52;  instead  of  seeing  seven  4's,  we  think 
seven  4's  are  28  and  5  are  33.  This  is  the  only  explana- 
tion necessary  when  the  tables  are  taught  by  the  first 
method  given  here. 

WHEN  THE  MULTIPLIER  HAS  Two  OB  MORE  FIGURES 

Before  written  multiplication  by  two-figured  multi- 
pliers is  taken  up,  the  pupil  should  understand  the  effect 
of  annexing  a  zero  to  a  whole  number.  He  should  never 
be  allowed  to  multiply  by  10  by  writing  down  the  multiplier 
beneath  the  multiplicand,  as  is  so  often  seen  even  in  our 
best  schools.  The  pupil  should  be  shown  that  annexing 


THE   PRIMARY   FACTS  OF   MULTIPLICATION      59 

a  zero  to  a  whole  number  has  the  effect  of  moving  each 
digit  one  order  higher,  and  thus  multiplies  each  digit  by 
10.  Thus  in  36,  by  annexing  a  zero  making  360,  the 
6  ones  become  6  tens,  and  the  3  tens  become  3  hundreds. 
If  place  value  has  not  been  taught  up  to  this  time,  it 
must  be  taught  here  if  the  pupil  is  to  see  why  annexing  a 
zero  multiplies  by  10.  This  may  be  followed  by  showing 
that  to  multiply  by  any  number  of  tens,  as  40,  first  multiply 
by  4  and  then  annex  a  zero  to  the  product,  which  multi- 
plies the  result  by  10.  These  two  facts  —  that  annexing 
a  zero  multiplies  an  integer  by  ten,  and  that  any  integer 
is  multiplied  by  a  multiple  of  ten,  by  multiplying  it  by 
the  multiple  and  annexing  a  zero  to  the  product  —  are 
preliminary  to  taking  up  two-figured  multipliers. 

Pupils  in  the  third  or  fourth  grade,  where  this  work  is 
first  taken  up,  are  too  young  to  follow  a  full  rationaliza- 
tion of  the  process.  In  order  that  the  pupil  may  satisfy 
his  curiosity  as  to  the  reason  for  performing  the  work 
as  he  does,  the  teacher  may  ask  some  such  questions  as 
the  following :  "  Three  8's  and  two  more  8's  are  how 
many  8's?"  "Six  5's  and  three  more  5's  are  how  many 
5's?"  Then,  "Five  83's  and  twenty  more  83's  are  how 
many83's?"  The  class  should  reply, "  Twenty-five  83 
83's."  The  teacher  may  then  say,  "We  will  find  25 
5X83  and  then  20X83  and  add  the  results,  and  415 
that  will  give  us  25X83."  As  the  work  is  per-  166 
formed  by  the  teacher  upon  the  blackboard,  the  2075 
pupils  giving  the  products  as  she  records  them,  she 
should  point  out  that  the  first  product  (415)  is  5X83. 
Now  in  finding  20X83,  the  teacher  gets  from  the  class 


60  THE   TEACHING  OF  ARITHMETIC 

that  20  is  10  times  2,  and  says,  "Then  if  I  multiply  83 
by  2  and  place  the  first  result  in  tens'  place,  that  will  be 
multiplying  the  result  by  10,  so  in  that  way  I  multiply 
by  10X2  or  20."  It  will  be  observed  that  this  is  a  very 
imperfect  rationalization,  but  it  will  be  found  that  it  is 
enough  of  the  "reason  why"  to  satisfy  the  curious  and 
is  perhaps  as  much  as  the  class  can  comprehend,  if  not 
more. 

As  in  all  cases,  the  work  is  first  presented  several  tunes 
by  the  teacher,  then  the  class  works  at  the  blackboard 
under  her  direct  supervision  until  the  process  is  mastered 
and  before  any  home  or  seat  work  is  given. 


CHAPTER  VII 

THE  PRIMARY  FACTS  AND  PROCESSES  OF 
DIVISION 

THE  TABLES 

THE  primary  facts  of  diyision  follow  at  once  from  the 
multiplication  fafets.  Thus,  ask  such  questions  as, 
"Thfee  5's  are  how  many?"  .(Answer,  15.)  "How  many 
5's  hi  15?"  "Four,6Ts  are  how  many?"  "How  many 
6's  in  24  ?  "  "  *Three  7ys-  are  how  many  ?  "  "  How  many 
7'sin21?"  • 

Notice  that  thege  are  the  questions  that  should  bring 
out  an  answer  from  the  child's  mental  picture  8 

of   multiplication.    Thus,    4X8   means   four    4vo_  8 
8's  written  and  added,  as  in  the  margin.    So,  8 

when  the  question,  "How  many  8's  in  32  ?"  is 
asked,  the  answer  very  naturally 'is  "Four."  32 

The  notations  should  then  be  given  by  saying,  "This 
is  the  way  we  write  the  fact  that  there  are  four 
8's  in  32."    As  a  drill  to  fix  the  meaning  of  the    8)32, 
notations,  have  a  large  miscellaneous  number  of         4 
the  facts  with  answers  written  as : 

3)21^      6)30       8)56       5)45       7)42       4)28       6)24 
7  5  7  9  6  7  4,  etc. 

61 


THE   TEACHING  OF   ARITHMETIC 

and  ask  the  class  to  read  them.  They  should  say,  "There 
are  seven  3's  in  21 " ;  "There  are  five  6's  in  30" ;  "There 
are  seven  8's  in  56";  etc.  Do  not  allow  the  expression 
"goes  into,"  so  often  heard,  and  so  often  pronounced, 
"gus-sin-tu." 

When  the  notation  is  understood,  have  the  pupils 
write  out  all  the  division  tables  in  this  way  from  their 
knowledge  of  multiplication. 

It  will  be  observed  that  the  meaning  of  the  division 
here  presented  is  measurement;  that  is,  the  quotient  shows 
how  many  times  the  dividend  contains  the  divisor.  When 
these  are  well  known,  the  partition  phase  of  division  may 
be  presented.  This  phase  is  shown  as  follows : 


5 

6 

8 

3 

2 

4 

5 

6 

8 

7 

6 

3 

2 

4 

5 

6 

8 

7 

6 

3 

2 

4 

5 

6 

8 

7 

6 

9 

6 

12 

20 

24 

32 

21 

18,  etc. 

Write  upon  the  blackboard  several  sets  of  equal  addends 
as  shown  above.  Now,  if  it  has  not  been  done  before, 
show  the  class,  dividing  up  some  object  as  you  do  so, 
that  one  of  the  three  equal  parts  of  anything  is  one  third 
of  it ;  one  of  the  four,  one  fourth  of  it ;  one  of  the  five, 
•me  fifth  of  it,  etc.,  and  thus  through  the  rhythm  of  it, 
they  can  at  once  tell  you  the  name  of  one  of  any  number 
of  equal  parts  of  anything. 

Now,  say,  "What  is  one  of  the  three  equal  numbers 
that  make  9?"  (Answer,  3.)  Then,  "What  part  of  9 
shall  we  call  3?"  (Answer,  one  third  of  it.)  "What  is 


THE   PRIMARY   FACTS  OF   DIVISION  63 

one  of  the  four  equal  numbers  that  make  20  ?  "  (Answer, 
5.)  "  What  then  shall  we  call  5  ?  "  (Answer,  one  fourth 
of  20.) 

These  questions  are  first  asked  about  a  large  number 
of  facts  written  upon  the  blackboard  in  addition  form. 
Later  they  are  asked  and  answered  from  the  mental 
picture  of  the  multiplication  facts. 

THE  NOTATION  OF  UNIT  FKACTIONS 

Without  any  further  discussion  of  fractions,  except 
the  limited  meaning  and  use  shown  above,  the  written 
notations  may  be  given : 

Thus,  one  sixth  is  written  £ 

one  seventh  is  written  y 
one  eighth  is  written  %,  etc. 

The  pupil  is  now  ready  to  write  the  facts  that  he  has  just 
given  orally  as  follows : 

i  of  6  =  2;       i  of  8=2;       £  of  15  =  3;       i  of  24  =  4,  etc. 

The  pupil  should  be  shown  that  the  first  form  of  ex- 
pressing division  may  be  interpreted  as  "a  part  "of " 

A\A 

when  the  divisor  is  an  abstract  number.    Thus,    —   may 

2         • 

be  interpreted  either  as  "There  are  two  3's  in  6,"  or 
"One  third  of  6  is  2." 

These  are  the  two  most  important  forms  of  expressing 
division,  but  for  the  purpose  of  brief  written  analysis 
the  form  6-5-3  =  2  should  be  given. 


64 


THE    TEACHING  OF   ARITHMETIC 


Since  the  form  for  written  work  is  the  first  one  given 
(  —  J,  this  is  the  form  that  should  be  used  in  all  sight 

drills.    It  may  be  interpreted  as  either  the  "measure- 
ment" or  the  "partition"  type  of  division. 

A  mere  knowledge  of  the  division  tables  is  not  sufficient 
for  successful  written  work.  Thus,  to  find  3)162,  the 
first  division  requires  not  one  of  the  primary  facts,  but 
the  fact  that  there  are  five  3's  and  1  in  16.  As  a  result, 
it  is  seen  that  not  only  is  drill  upon  the  tables  needed,  but 
the  pupil  must  drill  upon  every  possible  dividend  from 
one  times  the  divisor,  to  ten  times  the  divisor.  Thus, 
if  "the  sixes"  are  being  studied,  a  drill  is  needed  upon 
some  such  chart  as  the  following,  including  all  numbers 
from  6  to  60  inclusive : 

ABCDEFGHIJK 


1 

20 

38, 

60 

29 

7 

49 

42 

11 

36 

66 

24 

2 

17 

48 

6 

67 

34 

16 

23 

53 

66 

15 

60 

3 

40 

9 

"41 

10 

46 

37 

62 

33 

18 

59 

19 

4 

28 

39 

30 

61 

22 

26 

13 

63 

46 

12 

44 

5 

47 

21 

8 

27 

36 

14 

43 

31 

26 

64 

32 

In  drilling  upon  the  table,  for  example  in  column  B, 
the  pupil  says,  "  6  and  2  remaining ;  8 ;  1  and  3  remain- 
ing; 6  and  3  remaining;  3  and  3  remaining."  A  few 
minutes  daily  from  such  a  chart  will  greatly  aid  in  secur- 
ing rapid  and  accurate  written  work  in  short  division. 


THE   PRIMARY   FACTS  OF   DIVISION  65 

THE  FIRST  WRITTEN  DIVISION 

The  first  written  work  may  be  taken  up  along  with 
the  tables  and  should  proceed  very  gradually  from  the 
tables  themselves.  Thus,  when  the  pupil  writes 

— '    —     - — ,  etc.,  it  is  a  very  easy  step  to  get  him  to 
43        6 

2)86,  2)48,  2)128 
take  up  such  examples  as    '—-*    L—    '—— ,  etc.,  where 

T:O  ^TC  O4 

there  are  no  remainders  after  each  division.  The  oral 
drills  prepared  the  pupils  to  think  5-j-2  =  2  and  1  remain- 
ing; 9-i-2=4  and  1  remaining;  7-7-2  =  3  and  1  remaining, 
etc.  Therefore,  it  is  an  easy  step  to  show  that  the  1 
remaining,  used  with  the  number  that  follows,  is  the  nex£ 
number  to  divide. 

If  it  seems  wise  to  rationalize  the  work  more  fully, 
it  may  be  done  objectively.  For  example,  to  show  that 
52-f-2=26,  show  52  as  five  bundles  of  ten  splints  and 
two  ones.  Now,  if  the  five  tens  are  arranged  in  two  equal 
groups,  there  are  2  tens  in  each  group  and  1  ten  remaining. 
Now,  this  ten  and  2  make  12  ones.  Dividing  12  into 
two  equal  groups,  there  are  6  in  each.  So  there  are  2 
tens  and  6,  or  26,  in  each  of  the  two  parts. 

It  will  be  seen,  however,  that  this  explanation  is  really 
based  upon  the  "partition  idea"  of  division,  and  not  upon 
"measurement."  One  could  not  ask  children,  "How 
many  tunes  do  5  tens  contain  2?"  and  expect  an  answer. 
So,  if  the  work  is  to  be  presented  objectively,  more  oral 
drill  upon  the  partition  phase  of  the  subject  is  necessary. 
Thus,  hi  giving  5-J-2,  7-«-2,  9-7-2,  ll-v-2,  etc.,  the  pupil 


66  THE   TEACHING  OF   ARITHMETIC 

says,  "One  half  of  5  is  2  and  1  undivided,"  "One  half  of 
7  is  3  and  1  undivided,"  etc. 

In  drilling  upon  the  table  given  on  page  64  the  pupil 
thinks  "One  sixth  of  20  is  3  and  2  undivided";  "One 
sixth  of  17  is  2  and  5  undivided,"  etc. ;  and  not  "There 
are  three  6's  and  2  remaining  in  20 " ;  "  There  are  two 
6's  and  5  remaining  in  17,"  etc. 

It  seems  unwise,  however,  to  take  up  the  matter  of 
rationalization  with  pupils  of  the  third  grade  unless  they 
are  exceptional  children.  It  seems  better  to  begin  with 
very  easy  work  as  given  above,  and  to  show  how  to  do 
the  work  and  then,  as  the  work  becomes  easy  through 
drill,  to  progress  to  more  difficult  exercises. 

LONG  DIVISION 

The  pupil  has  been  performing  short  division  for  nearly 
a  year  before  taking  up  long  division.  By  writing  down 
all  the  work  that  was  done  mentally,  the  pupil  should 
be  shown  the  longer  form. 

Thus :  268 

9)2412  9)2412 

268  18^ 

61 

54 

72 
72 

As  this  is  shown,  question  the  pupil  as  to  what  he 
thought  in  short  division  and  write  it  down  in  long  division. 
Thus,  in  short  division  he  thinks,  "There  are  two  9's  in 


THE   PRIMARY   FACTS  OF   DIVISION  67 

24  and  6  remaining,  for  2X9  =  18,  and  24-18  =  6."    In 
long  division  this  is  written  down. 

The  only  reason  that  pupils  find  long  division  difficult 
is  because  they  have  trouble  in  estimating  the  quotient 
figures.  This,  then,  is  the  feature  upon  which  they  should 
be  drilled.  However,  as  a  preparation  for  long  division, 
it  is  well  to  give  a  few  lessons  in  which  the  divisors  are 
multiples  of  from  2  to  9  times  some  power  of  ten,  the 
work  to  be  done  by  dividing  both  dividend  and  divisor 
by  that  power  of  ten  and  by  using  short  division. 

Thus:       3000)24000  400)1700  etc. 

8  4;  100  rem. 

The  teacher  should  recognize  that  it  is  much  easier 
to  estimate  the  quotient  figures  from  some  divisors  than 
from  others  and  that  some  quotients  are  much  more 
easily  determined  than  others.  Thus,  the  easiest  divisors 
are  those  that  are  nearly  some  multiple  of  10  as  61,  71, 
81,  etc. ;  or  59,  69,  79,  etc.,  the  child  thinking  as  he  uses 
them,  "About  60,  about  70,  about  80,"  etc.  The  easiest 
quotients  to  determine  are  those  that  fall  about  midway 
between  two  consecutive  multiples  of  10,  as  65  (midway 
between  60  and  70),  74,  75,  76,  84,  85,  86,  54,  55, 
56,  etc. 

It  is  seen  then  that,  as  to  difficulty,  there  can  be  at 
least  three  grades  of  long  division  exercises.  The  divi- 
dends should  be  made  by  selecting  the  divisors  and  quo- 
tients wanted.  The  types  are  61X75;  59X69;  64X59. 
The  first  of  these  will  have  an  easy  divisor  (61)  and  an 
easy  quotient  (75);  the  second  an  easy  divisor  (59), 


68  THE   TEACHING  OF   ARITHMETIC 

but  a  hard  quotient  (69) ;    and  the  third,  both  a  hard 
divisor  (64)  and  a  hard  quotient  (59). 

76 


Thus,  to  find  6156-7-81,  as  shown  by  fol-  81)6156 

lowing  the  work  in  the  margin,  the  only  con-  567 

sideration    is    61-f-8  =  7    and    5    remaining;  486 

48-^8  =  6.  486 

In  4209  -5-  61,  as  may  be  observed,  the  consideration  is 
not  merely  42-7-6  =  7,  but  a  further  observation  69 

that  the  next  digit  (0)  of  the  dividend  does    61)4209 
not  contain  the  next  digit  (1)  of  the  divisor  7         366 
times,  and  hence  the  quotient  digit  cannot  be  549 

larger  than  6.  549 

While  in  2124-J-36,  the  pupil  who  makes  the  same  kind 
of  estimate  as  he  made  above  will  observe  that  21-r-3  =  7 
and  try  6.  But  he  must  be  shown  that,  before  59 

writing   down   6   in   the   quotient,   he   should    36)2124 
think  3X6  =  18;   21-18  =  3;   this  3  with  the  2          180 
makes  32;    now,   32  does  not  contain  6  six  324 

times,   hence   the   quotient   cannot   be   larger  324 

than  5. 

These  three  illustrations  should  make  it  clear  that 
there  are  at  least  three  grades  of  exercises  as  to  diffi- 
culty, even  in  exercises  containing  the  same  number  of 
figures. 

A  large  number  of  each  class  of  exercises  should  be 
worked  out  and  charts  prepared  as  follows : 


THE   PRIMARY   FACTS   OF   DIVISION  69 

Type  I, 


A 

B 

C 

D 

E 

F 

G 

H 

1 

2025 

2916 

3645 

4293 

3807 

4374 

6103 

5265 

>  -J-81 

2 

6885 

5994 

6966 

5427 

6804 

7695 

6075 

7614 

3 

3726 

6156 

7857 

5184 

4617 

7723 

7533 

5913 

TypeH 


4 

5551 

4819 

3111 

4941 

3599 

5978 

2562 

5429 

>  -4-61 

5 
6 

5002 

4758 

3782 

3538 

4209 

3172 

4148 

2379 

2501 

3721 

4392 

2928 

5612 

2989 

2867 

4331 

TypeHI 


7 
8 
9 

4788 

7068 

5544 

4104 

2052 

5548 

7372 

5092 

>  -^76 

4028 

6308 

3496 

5776 

4864 

2888 

6536 

6612 

2736 

5624 

7296 

2812 

2128 

4332 

3572 

3268 

In  type  I,  it  is  easy  to  estimate  the  quotient  figure,  for 
dividing  the  number  represented  by  the  first  two  figures 
of  the  dividend  by  the  first  number  in  the  divisor  always 
gives  the  correct  quotient  figure.  This  is  not  always  the 
case  in  type  II.  A  mental  multiplication,  however,  will 
show  whether  or  not  this  result  follows.  But  in  type  III, 
more  careful  consideration  is  required. 

There  are  two  uses  of  such  drill  charts :  (1)  They  may 
be  used  for  "sight  work"  in  which  the  pupil  will  give  the 
first  quotient  figure  only ;  and  (2)  they  may  be  used  for 
the  usual  written  work. 


CHAPTER  VIII 
THE  USE  OF   GAMES  IN  NUMBER  WORK 

IT  is  clearly  recognized  by  all  who  have  given  any 
thought  to  the  subject,  that  a  very  necessary  condition 
for  true  learning  is  that  the  process  be  self-actuated 
through  motive  or  interest.  If  the  thing  taught  a  child 
can  be  made  to  meet  his  needs  or  interests,  he  finds  but 
little  trouble  in  learning  it.  Hence,  a  problem  of  the 
teacher  is  to  find  a  use  to  the  child  for  the  number  facts 
that  she  must  teach  him.  A  number  problem  has  no\ 
great  interest  to  a  child  unless  the  answer  to  it  serves 
some  purpose  —  meets  some  personal  need  or  appeals 
to  his  curiosity.  The  range  of  problems  that  have  any 
real  interest  to  a  pupil  learning  the  primary  facts  is  so 
limited  that  they  are  not  sufficient  to  hold  the  interest 
long. 

Play  is  the  chief  occupation  of  most  children  of  the 
lower  grades,  hence  the  appeal  through  the  play  instinct 
—  games,  dramatized  vocations,  etc.  —  is  a  very  strong 
one,  and  may  be  used  to  very  great  advantage  in  the 
early  number  work. 

The  games  may  be  grouped  as  to  use  as :  (1)  school- 
room games,;  (2)  playground  games;  and  (3)  home 
games. 

79 


THE    USE  OF   GAMES  IN   NUMBER    WORK       71 

The  schoolroom  games  must  be  those  through  which 
the  teacher  may  get  a  maximum  amount  of  number 
combinations  before  the  class  with  as  little  of  the  time 
taken  up  by  the  game  as  possible.  The  games  must  be 
so  selected  as  to  have  all  pupils  silently  attending  to  the 
number  combinations  when  not  actually  reciting. 

The  school  ground  and  home  games  are  more  to  furnish 
a  motive  for  learning  the  work  presented  in  class  and, 
hence,  the  number  work  should  in  no  way  overshadow 
the  recreative  elements  of  the  game.  But  such  games 
serve  to  show  the  pupil  that  what  he  learns  in  school  may 
be  made  use  of  out  of  school ;  and,  by  turning  them  into 
"make-believe"  games,  they  may  be  used  in  the  class- 
room drills,  as  will  be  shown  in  the  games  that  follow. 

As  to  then*  nature,  both  the  in-school  and  out-of-school 
games  may  be  grouped  under  three  general  heads :  (1)  scor- 
ing game's;  (2)  imaginative  or  "make-believe"  games; 
and  (3)  games  of  motor  activity. 

SCHOOLROOM  GAMES 
SCORING  GAMES 

The  scoring  games  are  perhapB  mmt  familiar  to 
all.  They  include  Bean  Bag,  Rin|H  J  Ten  Pins,  and 
such  standard  games  as  may  be  founom  any  toy  shop. 
Yet,  as  found  in  the  toy  shop,  they  bring  in  but  few  of  the 
combinations. 

The  essentials  of  a  scoring  game  are :  (1)  that  so  little 
skill  is  required  to  score  that  every  "throw"  will  give  a 
number  for  a  combination;  and  (2)  that  the  targets  in- 


72 


THE   TEACHING  OF   ARITHMETIC 


elude  all  the  nine  digits.    Below  are  targets  that  may  be 
used  in  some  of  the  games  that  follow : 


/ 

7 

4 

5 

9 

6 

J 

8 

2 

There  are  many  tossing,  bowling,  and  shooting  games 
that  may  be  easily  devised  by  any  resourceful  teacher. 
The  tossing,  bowling,  or  shooting,  however,  does  not 
have  to  be  carried  on  to  any  great  extent  in  the  school- 
room. Any  of  the  following  scoring  games  may  be  used 
to  score  a  few  times,  then  the  game  turned  into  a  "  make- 
believe"  game.  Thus,  the  teacher  may  point  to  any 
number  on  such  targets  as  these  drawn  above  and  pre- 
tend that  they  are  the  scores  made  by  pupils,  thus  motivat- 
ing the  work  through  "make-believe"  target  practice. 

THROWING  THE  ARROW 

A  target  about  two  feet  across,  like  any  of  those  shown, 
may  be  made  0!aBt  board  and  covered  with  burlap  to 
hide  the  marks  Wm  the  shots,  and  hung  against  the 
wall.  .  The  pupils  throw  an  arrow  made  as  follows : 
take  a  large  pin  or  a  slim  finishing  nail  sharpened  to  a 
point,  a  small  cork  about  one  inch  long  and  three  fourths 
of  an  inch  in  diameter,  and  three  feathers  from  five  to 
seven  inches  long,  and  make  an  arrow  as  shown  in  the 
picture.  This  can  be  thrown  with  great  accuracy. 


THE    USE  OF  GAMES  IN  NUMBER   WORK       73 

Let  each  child  throw  twice  and  have  the  sum  or  product, 
depending  upon  the  drill,  of  the  numbers  "hit  if  he  calls 
them  correctly.  After  a  few  throws,  the  .teaCher  may 
merely  point  to  two  numbers 
on  the  target,  asking  a  pupil 
to  name  the  score.  If  correct, 
it  is  added  to  his  other  scores. 
After  a  pupil  can  perform 
written  addition,  he  may  add  his  scores  every  five  turns 
or  chances  and  thus  the  same  game  may  motivate  the 
written  addition. 

BEAN  BAG 

Take  a  large  cardboard,  or  a  light  board,  and  make 
nine  holes  in  it,  numbering  them  from  1  to  9  inclusive. 
Leaning  it  against  a  support  of  some  kind,  the  children 
toss  small  bean  bags  through  the  holes,  scoring  as  hi  the 
arrow  game. 

HOOK  IT 

A  board  with  nine  hooks  on  it,  numbered  from  1  to  9 
inclusive,  is  hung  against  the  wall.  The  children  try  to 
ring  the  hooks  by  tossing  soft  rubber  rings. 

BRIDGE-BOARD  MARBLE  GAME 

Cut  nine  semicircular  archways  about  three  inches  in 
diameter  from  one  edge  of  a  board.  Then  support  the 
board  edgewise  on  the  floor  so  that  these  arches  are  next 
the  floor.  Number  the  archways  from  1  to  9  and  let 
the  children  roll  marbles  through  the  openings,  scoring 
as  in  the  preceding  games. 


74  THE   TEACHING   OF   ARITHMETIC 

BOWLING  WITH  MARBLES 

A  target  may  be  marked  off  on  the  floor  and  a  marble 
rolled  into  it,  the  child  scoring  the  numbers  upon  which 
the  marble  stops. 

These  are  but  a  few  suggestions  as  to  the  possibilities 
with  scoring  games.  The  methods  of  scoring  may  easily 
be  varied. 

IMAGINATIVE   OR  "MAKE-BELIEVE"    GAMES 

These  include  the  dramatization  of  various  vocations, 
as  "playing  store,"  "the  mailman,"  "the  milkman," 
and  also  the  pretended  playing  of  some  scoring  game,  or 
other  game,  by  use  of  diagrams  drawn  upon  the  black- 
board. 

There  is  no  end  to  imaginative  games  that  may  be 
made  by  the  teacher.  The  following  will  show  how  a  re- 
sourceful teacher  used  a  game,  which  she  called  "  Playing 
Fireman,"  for  several  weeks  without  any  loss  of  interest, 
and  how  the  game  suggested  itself  to  her.  The  fire  de- 
partment had  just  passed  the  school  ground  during  inter- 
mission and  the  children  were  greatly  interested  in 
watching  it.  When  the  period  for  number  work  came, 
the  teacher  asked  how  many  would  like  to  play  "Fire- 
man." Of  course,  all  wanted  to  play.  She  stepped  to 
the  blackboard  and  drew  two  ladders  leading  to  the  top 
story  of  a  burning  building.  Between  the  rungs  of  the 
ladders  she  placed  the  number  combinations  upon  which 
she  wished  to  drill.  She  said,  "Now  we  will  play  that 


THE    USE   OF   GAMES  IN   NUMBER    WORK.      75 

the  one  who  can  run  up  one  ladder  and  down  the  other 
without  making  an  error  has  rescued  some  one  from  the 
burning  building;  but  those  who  make  a  mistake  have 
fallen  down  and  so  are  not  good  firemen." 

After  a  delightful  period  of  "Playing  Fireman"  it 
was  suggested  that  two  permanent  companies  should 
be  organized  and  that  each  company  should  have  a  fire 
chief.  The  pupils  said  that  the  strongest  and  boldest  — 
quickest  and  most  accurate  —  should  be  chosen  chief ; 
but,  of  course,  each  child  hoped  that  he  might  be  the  one 
chosen  for  that  office.  Then  followed  more  drill  for  a 
few  days  before  having  a  contest  to  determine  who  should 
be  the  chiefs.  At  last,  two  were  chosen.  The  chiefs 
wanted  to  choose  then:  own  companies.  As  a  result, 
more  drill  was  needed  in  order  that  the  chiefs  might  see 
whom  they  wished  to  choose.  The  first  day  resulted  in 
choosing  but  two  or  three  for  each  side.  Finally,  all 
were  chosen.  The  next  step  was  for  each  chief  to  train 
his  men  for  a  special  exhibition  of  skill.  A  day  was  fixed 
for  the  exhibition.  Then  each  man  in  each  company 
gave  all  the  combinations.  The  time  required  and  the 
errors  made  were  noted  in  order  to  tell  which  company 
won  the  prize. 

Then  there  was  a  big  fire  in  a  down-town  district  and 
whole  blocks  were  aflame  and  many  ladders  were  put  up 
to  rescue  people.  Each  team  was  greatly  interested  in 
seeing  who  could  save  the  most  people  in  a  given 
time. 

This  is  discussed  in  detail  as  being  somewhat  typical 
of  what  may  be  done  with  many  such  number  games. 


76  '  THE   TEACHING  OF  ARITHMETIC 

.SIMON  SAYS  "THUMBS  UP" 

It  is  often  desirable  to  have  a  single  pupil  recite  all  of 
a  group  of  facts.  Thus,  a  teacher  may  wish  to  have  a 
pupil  give  several  or  all  of  the  facts  of  addition  or  multi- 
plication. But,  unless  she  can  in  some  way  get  the 
attention  of  the  whole  class  and  get  all  to  attend  silently 
to  the  combinations,  the  time  of  all  but  the  one  reciting 
is  wasted.  The  following  game  shows  how  a  resourceful 
teacher  got  the  attention  of  the  whole  class  in  such  a  drill 
without  any  loss  of  time  and  without  disturbance  or  extra 
material. 

She  called  the  game,  "  Simon  Says  *  Thumbs  Up.' '''  One 
pupil  stood  before  the  class  and  showed  the  number  cards. 
Another  stood  and  gave  the  combinations.  The  class 
sat  with  "thumbs  up."  When  a  mistake  was  made,  all 
thumbs  were  to  turn  down.  The  last  to  turn  thumbs 
down  stood  to  give  the  combinations,  and  the  first  to  do 
so  had  the  privilege  of  showing  the  cards. 

SURPRISE  PARTY 

In  using  toy  money  and  making  change  a  good  plan 
is  to  have  a  "Surprise  Party."  Ask  the  children  how 
many  would  like  to  go  to  a  surprise  party.  Of  course, 
all  want  to  go.  Then  ask  what  each  would  like  to  take 
and  write  his  answer  on  the  board,  as  apples,  bananas, 
oranges,  candy,  nuts,  etc.  Next,  appoint  a  clerk  to  step 
to  the  board  and  write  the  prices  at  which  the  articles 
can  be  bought  at  the  store.  Give  the  children  toy  money 
and  let  them  buy  the  things  of  the  clerk.  Beads,  blocks, 


THE   USE  OF  GAMES  IN  NUMBER   WORK      77 

chalk,  etc.,  may  represent  the  articles.  If  the  clerk  makes 
a  mistake,  discharge  him  and  let  some  one  else  take  his 
place. 

PLAYING  STORE 

This  game  may  be  adapted  to  any  of  the  primary  grades 
and  resembles  somewhat  the  "Surprise  Party."  A  table 
at  the  front  of  the  room  may  serve  as  a  counter  of  the 
store.  Let  the  children  determine  the  nature  of  the 
things  to  be  bought  and  sold — fruit,  flowers,  toys,  sport- 
ing goods,  furniture,  groceries,  dolls,  or  candy.  If  varied 
frequently,  the  interest  is  keener.  Things  made  in 
manual  training  or  handwork  periods  may  be  used  to 
form  the  stock.  Charts  may  be  used  in  place  of  actual 
objects.  Added  interest  is  gained  by  having  toy  money, 
toy  scales,  and  a  toy  telephone.  The  children  ask  the 
prices  of  what  they  want,  then  determine  how  much  theyx 
will  buy.  They  should  calculate  the  amount  of  their 
purchases  so  as  to  be  able  to  detect  errors  on  the  part  of 
the  storekeeper. 

LUNCH  ROOM 

One  child  is  the  keeper  of  a  make-believe  lunch  room. 
The  other  children  pretend  to  buy  their  lunches  from  him. 
A  chart  showing  the  cost  of  each  article  of  food  should  be 
posted. 

SOLDIERS 

The  children  form  a  company  and  choose  their  captain. 
The  captain  forms  his  men  into  ranks  and  files.  He 
then  drills  them  with  number  combinations.  The  answers 
should  be  given  promptly,  accurately,  and  with  decision, 


78  THE    TEACHING  OF  ARITHMETIC 

as  becomes  a  soldier.  Should  they  be  given  lazily  or 
incorrectly,  that  soldier  loses  his  rank,  or  may  even  be 
sent  to  barracks  (out  of  the  game).  Sometimes  two 
companies  may  combat  with  each  other,  the  company 
giving  most  combinations  correctly  being  victorious  and 
taking  the  others  captive  or  capturing  their  standard. 

KING  OF  THE  CASTLE 

A  child,  the  king,  sits  on  a  chair  in  the  front  of  the 
room.  The  other  children  try  to  dethrone  the  king  by 
giving  him  number  combinations  which  he  cannot  answer. 
As  long  as  he  gives  correct  answers,  he  remains  on  his 
throne  in  the  castle ;  but,  when  he  fails,  the  child  who  gave 
the  combination  becomes  king  in  his  place. 

TOM  TIDDLER'S  GROUND 

A  square  on  the  floor  represents  "Tom  Tiddler's 
Ground,"  in  the  center  of  which  Tom  is  drawn.  Some 
one  asks,  "  Who  will  dare  to  go  on  Tom  Tiddler's  ground  ?  " 
A  boy  in  the  class  stands  in  front  of  the  room  to  repre- 
sent Tom.  The  child  who  dares  to  go  on  Tom  Tiddler's 
ground  is  then  questioned  by  Tom,  who  asks  various 
number  combinations.  If  correct  answers  are  given 
three  times,  the  child  may  stay  on  his  ground  without 
flanger;  but  if  he  misses,  he  is  chased  out  by  Tom  and 
some  one  else  tries. 

HORSE  RACE 

Lay  before  each  child  a  flash  card,  as  6+4.  Give  a 
quick  review  to  be  sure  each  child  knows  his  own  card 


THE    USE  OF   GAMES  IN   NUMBER   WORK       79 

correctly.  Choose  as  many  children  for  horses  as  there 
are  aisles.  The  number  cards  are  the  hurdles,  and  the 
horses  all  start  together.  The  horses  giving  correct 
combinations  have  surmounted  the  hurdles  and  pass  on 
down  the  aisle.  If  a  horse  makes  a  mistake,  the  child 
holding  that  combination  stretches  his  hand  across  the 
aisle.  This  barrier  cannot  be  passed  until  the  correct 
combination  is  given.  The  first  horse  to  give  all  the  com- 
binations correctly  in  his  aisle  wins  the  race. 

ELEVATORS 

Write  a  series  of  combinations  as  in  the  margin.  4 
To  go  up,  add.  To  come  down,  subtract.  Each  2 
combination  represents  a  Jbor.  If  an  incorrect  3 
answer  is  given,  the  elevator  stops.  2 

5 

2 
JACK  AND  THE  BEAN  STALK 

After  telling  the  story  say  to  the  children,  "Let  us 
play  that  we  are  going  to  climb  the  bean-stalk  ladder  that 
Jack  climbed,  only  we  are  going  to  climb  it  in  a  different 
way."  Make  a  ladder  of  cardboard  so  that  all  the  class 
can  easily  see  it,  or  draw  one  on  the  board.  In  the  spaces 
write  number  combinations  in  addition,  subtraction,  or 
multiplication.  The  one  who  can  go  up  and  down 
the  ladder  most  rapidly  (by  giving  the  answers  to  the 
combinations)  and  without  stumbling  or  slipping  (making 
a  mistake)  can  be  Jack.  The  one  who  makes  no  mis- 
takes and  takes  the  shortest  time  to  go  up  and  down  the 


80  THE   TEACHING  OF  ARITHMETIC 

ladder  makes  the  best  Jack.  A  hard  sum  to  overcome 
may  be  given  for  the  giant  and  the  one  who  can  conquer 
this  big  giant  is  really  "  Jack-the-Giant-Killer." 

PICKING  APPLES 

Draw  an  apple  tree  on  the  board  with  apples  on  it. 
Each  apple  has  on  its  side  a  number  combination.  The 
children  pick  apples  by  giving  the  combinations.  The 
one  who  picks  the  most  apples  wins  the  game. 

SAVING  THE  CHEW 

A  sinking  ship  is  drawn  on  the  board.  Half  of  the  deck 
of  the  ship  is  in  the  water.  Combinations  as  8+2,  2X7, 
8—2,  10+2,  etc.,  are  placed  on  the  mast  and  on  every 
part  of  the  ship  which  is  above  water,  to  represent  the 
crew  of  the  ship  waiting  to  be  rescued.  Then  as  many 
boats  as  there  are  pupils  are  drawn.  Each  child  owns  a 
boat  and  his  initial  is  placed  on  it.  Then,  the  one  who 
saves  the  most  of  the  crew  —  that  is,  tells  the  greatest 
number  of  combinations  —  is  captain.  For  each  man 
rescued,  the  child  gets  a  flag  which  he  places  in  his  boat. 

PLAYING  SOLDIEES 

Divide  the  class  into  two  armies,  A  and  B.  Let  them 
choose  captains.  The  captain  of  army  A  asks  a  soldier 
of  army  B  a  combination.  If  the  soldier  cannot  reply 
correctly,  army  B  loses  a  soldier  and  army  A  gets  him. 
If  he  replies  correctly,  however,  he  has  the  privilege  of 
asking  a  soldier  of  army  A  a  combination.  The  army 


THE   USE  OF  GAMES  IN  NUMBER   WORK      81 


having  the  most  soldiers  at  the  end  of  a  stated  tune  wins 
the  battle.  This  game  may  be  used  in  subtraction, 
addition,  multiplication,  or  division. 

A  HOME-RUN 

Divide  the  class  into  two  baseball  teams.  Draw  a 
baseball  diamond  upon  the  blackboard  and  write  three 
or  four  numbers  at  each  base,  as  in  the  figure.  Some 
number  to  be  added  or  to  be 
used  as  a  multiplier  is  written 
in  the  pitcher's  place.  The 
pupil  makes  a  home-run  by 
giving  all  the  answers  (sums 
or  products)  as  the  teacher, 
starting  from  the  home  plate,  fi 
points  to  some  number  at 
each  base.  Thus,  to  use  the 
game  to  multiply  by  8,  the 
teacher  may  point  to  6,  9,  7, 
and  5,  the  pupil  calling  48, 
72,  56,  40,  which  constitutes  a  home-run.  If  a  mistake 
is  made,  say  at  the  second  base,  the  umpire  calls  "out 
on  second."  The  game  is  to  see  which  team  gets  the 
most  home-runs. 

A  GTJESSING  GAME 

L 

As  a  drill  in  addition,  the  teacher  may  say,  "I  am 
thinking  of  two  numbers  which  added  make  10."  The 
pupils  guess  all  of  the  combinations  of  numbers  which 
added  make  10,  and  when  the  last  combination  is  given, 


82  THE   TEACHING  OF  ARITHMETIC 

the  teacher  may  call  that  the  right  one.  The  same  drill 
may  be  used  in  multiplication.  For  example,  the  teacher 
may  say,  "I  am  thinking  of  two  numbers  whose  product 
is  24."  The  pupils  will  guess  two  12's,  three  8's,  six  4's. 
Or,  the  teacher  may  say,  "I  am  thinking  of  the  'nine 
times'  table."  A  child  will  ask,  "Are  you  thinking  of 
81?"  The  teacher  will  say,  "No,  I  am  not  thinking  of 
9X9."  When  the  correct  answer  is  given,  the  guesser 
may  suggest  the  table  and  answer  the  questions,  and 
thus  it  may  be  carried  out  between  two  children,  the 
teacher  acting  as  umpire. 

STORY  GAMES 

The  stories  with  which  the  children  are  familiar,  such 
as  "Clytie,"  "Snowwhite,"  "Sleeping  Beauty,"  etc., 
may  be  utilized  in  addition,  subtraction,  and  multiplica- 
tion drills.  For  example,  in  "Sleeping  Beauty"  a  picture 
of  the  schoolhouse,  thorns,  huge  walls,  and  a  castle  may 
be  drawn  on  the  board.  Several  combinations  may  be 
placed  on  each  and  a  boy  may  start  for  jthe  castle.  If  he 
succeeds  in  getting  into  the  castle,  having  overcome  all 
obstacles  by  giving  the  correct  combinations  on  the  various 
obstacles,  he  is  dubbed  knight  and  some  emblem  is  pinned 
upon  the  lapel  of  his  coat  to  signify  as  much.  If  he  fails, 
some  one  else  may  try.  Competition  may  be  gained  by 
having  several  such  series  on  various  boards,  and  several 
children  trying  to  gain  the  castle  first.  The  one  who 
gives  the  combinations  soonest  is,  of  course,  the  honored 
one. 

The  children  may  take  a  ride  with  Clytie  in  her  wonder- 


THE    USE  OF   GAMES  IN   NUMBER    WORK       83 

ful  seashell  carriage,  or  visit  the  seven  dwarfs  in  their 
mountains  of  gold.  If  so,  the  dwarfs  or  mermaids  re- 
spectively may  hold  large  cards  on  which  are  combinations, 
and  these  must  be  given  before  the  child  is  allowed  to 
arrive  and  visit  the  dwarfs'  cave  or  Clytie's  beautiful 
home.  The  children  are  enthusiastic  over  these  simple 
devices. 

These  examples  will  serve  to  show  the  possibilities 
along  this  line  of  schoolroom  games.  A  teacher  inter- 
ested in  devising  games  to  motivate  her  work  will  find 
suggestions  on  every  hand. 

GAMES  OF  MOTOR  ACTIVITY 

In  schoolrooms  devoted  to  a  single  grade,  so  that 
moving  about  does  not  disturb  others,  there  are  many 
games  in  which  there  is  some  movement ;  yet  the  amount 
of  number  work  involved  is  great  enough  to  meet  the 
requirements  of  the  classroom. 

O-U-T  SPELLS  OUT 

The  pupils  may  be  arranged  in  a  circle  or  remain  in 
their  seats.  The  first  child  gives  some  combination,  as 
"5  and  7."  The  next  in  turn  gives  the  sum  and  some 
other  number,  as  "12  and  3."  The  next  may  say,  "15 
and  8."  The  game  continues  in  this  way,  the  first  num- 
ber given  each  tiDaebfiiftg^tbe,8um  of  the  last  two  numbers 
given.  For  the  fist  mistake 'a  child  makes  he  scores  the 
letter  O ;  for  the  second,  It ;  and  for  the  third,  T.  He  is 
then  out  of  the  game. 


84  THE   TEACHING  OF  ARITHMETIC 

BASKETBALL 

Large  numbers  are  pinned  on  each  child.  The  chil- 
dren stand  in  a  circle  about  one  child  in  the  center  who 
tosses  a  large  ball  to  some  one  in  the  circle,  at  the  same 
time  calling  out  some  number.  If  the  child  in  the  circle 
catches  the  ball  and  gives  the  correct  sum,  he  exchanges 
places  with  the  one  in  the  center  and  is  privileged  to  toss 
the  ball.  If  he  fails  to  catch  the  ball  or  gives  the  wrong 
sum,  he  remains  in  the  circle. 

A  COLUMN  RELAY  RACE 

Divide  the  class  into  two  teams.  Give  each  a  large 
card  containing  a  number  less  than  ten.  The  children 
stand  in  line.  The  leader  of  each  team  calls  out  his  num- 
ber and  shows  his  card.  The  next  in  turn  shows  his  num- 
ber and  adds  it  to  the  first,  giving  the  sum.  The  next 
shows  his  number  and  adds  it  to  the  sum  already  given, 
and  so  on  to  the  end  of  the  line.  The  line  completing 
the  addition  first  wins  the  game  if  the  sum  is  correct. 

The  numbers  of  one  team  should  duplicate  those  of 
the  other  but  should  be  arranged  in  different  order.  The 
sums  will  then  be  the  same. 

A  NUMBER  RACE 

Divide  the  class  into  two  teams  —  The  Reds  and  The 
Blues.  Write  several  numbers  not  larger  than  eighteen 
upon  the  blackboard,  as  12,  17,  15,  13,  and  18.  Let  the 
leader  of  each  team  go  to  the  board.  The  remaining 


THE    USE   OF   GAMES  IN  NUMBER    WORK       85 

pupils  in  turn,  alternating  from  one  side  to  the  other, 
call  some  combination,  the  sum  of  which  is  on  the  board. 
Thus,  a  pupil  from  the  Blues  calls  "7  and  8."  The  two 
pupils  at  the  board  see  who  can  first  cross  out  15.  After 
each  has  called  a  combination,  the  scores  are  taken.  If 
the  leader  from  the  Reds  has  won  8  tunes  and  the  one 
from  the  Blues  has  won  6  tunes,  the  score  is  2  for  the 
Reds.  If  a  pupil  calls  a  combination  whose  sum  is  not 
on  the  board,  that  takes  one  from  the  score  of  his  side. 

A  BLACKBOARD  RELAY  RACE 

Before  the  class  meets,  the  teacher  places  all  the  com- 
binations to  be  drilled  upon  on  the  board,  beginning  at 
the  ends  of  the  board  and  working  toward  the  center,  at 
which  a  goal  is  drawn.  The  same  combinations  are  on 
either  side  of  the  goal  but  written  in  a  different  order. 
The  class  is  divided  into  two  teams.  At  a  given  signal, 
the  leader  of  each  team  runs  to  the  board  and  records  the 
result  of  the  first  combination,  returns  and  gives  his  chalk 
to  the  next  in  line,  who  records  the  next  result,  giving 
his  chalk  to  the  next  in  order,  and  so  on  until  the  goal  is 
reached. 

OLD  WITCH 

Have  an  old  witch,  mother,  and  children.  Give  the 
children  numbers  such  as  15,  14,  16,  etc.  The  old  witch 
comes  to  the  door  and  says,  "I  want  a  child."  "Is it 
8  and  7?"  asks  the  mother.  "No,  it  is  not  15,"  says  the 
witch.  "Is  it  7  and  7?"  asks  the  mother.  "Yes,  it  is 
14,"  says  the  witch.  "You  must  catch  me  then,"  says 


86  THE   TEACHING  OF  ARITHMETIC 

14,  and  he  runs.  If  the  old  witch  catches  him,  he  is  hers ; 
if  not,  he  is  free.  If  the  child  does  not  know  the  combina- 
tion and  is  not  ready  to  run,  he  belongs  to  the  old  witch. 
The  teacher  may  take  the  place  of  the  mother,  so  that 
new  combinations  may  be  given.  Addition,  subtraction, 
and  multiplication  combinations  may  be  used. 

SHEPHERD  AND  WOLF 

One  child  is  the  shepherd  and  one  is  the  wolf,  and  all 
the  rest  are  sheep.  The  shepherd  stands  at  one  "end  of 
the  room,  the  sheep  at  the  other,  and  the  wolf  between 
the  two.  The  shepherd  has  some  number,  as  5.  The 
sheep  each  have  numbers  as  1,  2,  3,  4,  5,  etc.  The  wolf 
gives  a  number,  as  12.  When  the  wolf  says  12,  the  num- 
ber that,  added  to  the  shepherd's  number  5,  will  make 
12  —  i.e.  7  —  must  run  across  the  room.  If  the  wolf 
says  8,  then  number  3  runs,  etc.  If  a  sheep  fails  to  run 
when  he  should,  he  belongs  to  the  wolf,  who  takes  him  to 
his  den.  If  he  runs  and  is  not  caught,  he  may  go  back 
to  his  former  place. 

BEAST,  BIRD,  AND  FISHES 

Children  stand  in  the  circle  with  one  child  in  the 
center.  The  one  in  the  center  points  quickly  to  a  child 
in  the  circle,  giving  him  a  combination  to  answer.  Then 
the  child  in  the  center  counts  1,  2,  3,  4,  5.  The  one 
pointed  to  must  give  the  answer  before  5  is  counted  or 
he  is  "out"  of  the  game. 


THE   USE  OF  GAMES  IN  NUMBER   WORK      87 

WHO  TAPS 

One  child  is  outside  of  a  circle  of  children.  He  taps 
a  child  on  the  back.  "Who  taps?"  demands  the  one 
tapped.  The  answer  given  is  a  combination.  Then  the 
other  must  reply  with  the  answer  to  the  combination  or 
else  go  outside  of  the  circle. 

CATCH  THE  BALL 

The  children  form  two  lines.  Each  has  a  number 
pinned  on  in  full  view.  Some  one  tosses  a  ball,  or  a  bean 
bag,  to  some  one  on  the  other  side,  who  gives  the  sum 
(product  or  difference)  of  his  number  and  that  of  the 
pitcher.  If  correct,  he  tosses  it  back  to  some  one  on  the 
first  side,  who  answers  in  the  same  way.  When  a  mis- 
take is  made,  the  one  making  it  drops  out  of  the  game 
and  the  ball  or  bag  goes  to  the  head  of  the  opposite  side 
to  toss. 

NUMBER  BASKET  UPSET 

The  children  sit  in  a  circle  with  numbers  pinned  on 
them.  Some  one  stands  in  the  center  and  gives  a  com- 
bination three  times  in  succession,  as  "7  and  2,  7  and  2, 
7  and  2."  The  child  with  9  must  say  "9"  before  the 
one  in  the  center  has  finished.  If  he  is  not  quick  enough, 
he  forfeits  his  seat  to  the  child  in  the  center  and  then 
gives  the  combinations.  The  one  in  the  center  may  at 
any  time  say,  "Number  basket  upset,"  and  every  child 
must  change  his  seat,  the  one  in  the  center  also  trying 
to  get  a  seat.  There  will  be  one  child  left  without  a  chair 
who  must  be  in  the  center  and  give  the  combinations. 


88  THE   TEACHING  OF  ARITHMETIC 

GRUNT 

Each  child  is  given  a  number,  and  all  but  one  form  a 
circle.  One  child  in  the  center  is  blindfolded.  He  has 
a  pointer  or  small  rod.  The  children  march  around  him 
in  a  circle.  When  the  one  in  the  center  taps  the  floor 
with  the  rod,  all  stop.  He  touches  some  one  in  the  circle, 
who  takes  one  end  of  the  rod  and  gives  the  sum  of  his 
number  and  the  number  of  the  one  who  is  blindfolded. 
The  one  blindfolded  then  "guesses"  the  number  of  the 
one  touched,  which  he  does,  of  course,  by  subtracting 
his  number  from  the  sum  given.  If  correct,  the  one 
touched  is  blindfolded. 

A  JUMPING  GAME 

Two  children  "turn"  a  rope  and  the  others  stand  in 
line  ready  to  jump  the  rope  in  turn.  Each  child  is  to 
jump  as  many  times  as  he  can,  counting  by  some  number, 
as  5,  each  tune  he  jumps.  Whoever  counts  farthest  wins. 
If  he  makes  a  mistake  in  counting,  he  stops  jumping. 
This  gives  practice  in  adding  equal  addends. 

BALL  AND  HOOP 

A  hoop  is  suspended  in  the  room.  The  object  of  the 
game  is  to  throw  a  ball  through  the  hoop.  The  players 
are  divided  into  sides  and  take  turns  alternately.  Each 
time  a  child  throws  the  ball  through  the  ring,  the  captain 
of  the  opposing  team  challenges  him  with  some  number 
combination.  If  he  can  answer  it,  he  scores  two  for  his 


THE    USE   OF  GAMES   IN   NUMBER   WORK       89 

team ;  but,  if  he  cannot  answer  it  correctly,  he  loses  one 
point  for  his  team. 

HOT  BUTTER  BEANS 

Instead  of  hunting  for  a  bean,  as  in  the  real  game, 
let  the  children  hunt  for  a  numbered  block.  Let  one 
child  hide  the  block.  When  ready,  the  others  hunt. 
The  hider  says  "hot"  or  "cold"  according  to  the  near- 
ness of  the  children  to  the  block.  The  child  who  finds 
the  block  adds  the  number  to  his  score  if  he  can  call  the 
combination  quickly.  He  then  hides  a  block.  The 
number  on  the  block  is  changed  each  time.  Let  the 
final  score  be  100,  and  the  child  which  gets  it  first  wins 
the  game. 

FENCING 

The  children  stand  in  two  lines  facing  each  other. 
Each  child  must  be  opposite  an  opponent.  Then  some 
number  is  decided  upon,  as  twelve.  One  child  gives  a 
number,  such  as  seven,  and  his  opponent  is  required  to 
give  the  number  which  must  be  added  to  make  twelve. 
This  is  done  all  along  the  line.  Whenever  a  child  misses, 
the  mistake  is  recorded  against  him.  The  mistakes  are 
counted  after  a  given  time,  and  the  side  which  has  made 
the  fewest  mistakes  wins. 

OUT-OF-SCHOOL   GAMES 

Most  of  the  games  played  by  children  out  of  school 
may  be  adapted  to  number  games  without  taking  away 
the  recreative  elements  of  the  games.  These  games 


90  THE   TEACHING  OF  ARITHMETIC 

should  not  be  given  as  a  requirement,  nor  should  they 
involve  enough  number  work  to  task  the  children's 
abilities.  The  purpose  of  them  is  not  drill  in  number 
work,  but  they  are  given  to  motivate  the  classroom  work 
by  showing  a  use  of  the  things  they  are  learning.  The 
following  will  suggest  ways  of  using  the  well-known  games. 

Fox  AND  GopSE 

Large  numbers  are  pinned  on  each  child.  Choose  one 
child  for  the  fox  and  one  for  a  goose.  The  other  children 
stand  in  pairs  about  the  playground  with  linked  arms. 
The  fox  tries  to  catch  the  goose.  In  order  to  escape  from 
the  fox,  the  goose  may  link  arms  with  any  one  of  any 
couple,  first  calling  out  the  sum  of  the  numbers  of  the 
couple.  The  pupil  with  whom  the  goose  did  not  link 
arms  then  becomes  goose.  If  the  goose  calls  the  wrong 
sum,  she  is  not  safe  and  must  run  to  another  couple  and 
try  again,  or  be  caught. 

PUSSY  WANTS  A  CORNER 

All  children  are  numbered  with  large  visible  numbers, 
each  two  children  having  the  same  number,  and  all  but 
one  has  a  corner  (a  base  of  some  sort).  The  child  with- 
out a  corner  goes  from  one  to  another  saying,  "Pussy 
wants  a  corner."  The  one  to  whom  it  is  said  asks,  "What 
corner?"  The  reply  is  some  two  numbers  whose  sum 
(or  product)  is  represented  by  the  children.  The  two 
children  having  the  sum  (or  product)  exchange  places. 
The  child  without  a  corner  tries  to  get  one  of  the  vacant 
places.  Thus,  if  he  answers  "7  and  8,"  the  two  15's 


THE    USE   OF  GAMES  IN   NUMBER   WORK       91 

exchange  places.    If  he  can  get  one  of  the  places,  the  one 
left  becomes  "pussy." 

SQUAT  TAG 

Pin  a  visible  number  on  each  child.  Let  one  child 
be  chosen  as  "it."  He  chases  some  one  of  the  other 
children.  To  prevent  being  caught,  the  one  who  is 
chased  squats  down,  giving  the  sum  (or  product)  of  the 
number  pinned  on  "it"  and  his  own  number.  If  the  sum 
(or  product)  is  incorrect,  he  may  be  caught  and  then 
becomes  "it." 

Fox  A,NB  GEESE 

Divide  the  class  into  foxes  and  geese.  Give  the  foxes 
a  certain  amount  of  space  and  geese  the  same  amount. 
Geese  fly  over  into  the  territory  occupied  by  the  foxes. 
If  a  goose  is  caught  there,  she  must  answer  a  combination. 
If  she  answers  correctly,  she  may  go  free;  if  not,  she 
must  be  a  fox.  Foxes  may  visit  the  home  of  geese  and 
take  away  with  them  all  the  geese  which  cannot  answer 
correctly  the  combinations  put  to  them.  The  aim  of  the 
geese  is  not  to  be  caught;  or,  if  caught,  to  answer  cor- 
rectly the  combination  given  so  that  they  may  go  free. 

TAP  THE  RABBIT 

All  the  children  but  one,  who  is  to  be  the  leader,  form 
in  a  ring.  Each  child  has  a  number  pinned  upon  him. 
The  leader  runs  around  the  ring  and  taps  some  one  in 
the  ring  and  calls  a  number.  The  one  tapped  gives  the 
combination  of  his  number  with  the  number  called,  before 
he  starts  around  the  ring  in  the  opposite  direction.  The 


92  THE   TEACHING  OF  ARITHMETIC 

one  reaching  the  vacant  place  first  stays  in  the  ring ;  the 
other  is  leader.  This  game  may  be  played  with  addition, 
multiplication,  or  subtraction  combinations. 

THREE  DEEP 

Have  two  rings  of  children,  one  child  in  front  of  the 
other.  Have  two  children  outside  of  the  ring,  one  pur- 
sued by  the  other.  All  have  numbers  pinned  on  front 
and  back  of  them.  The  one  pursued  runs  and  stands  in 
front  of  a  child  in  the  ring,  who  quickly  gives  the  sum  of 
his  number  and  the  number  of  the  child  who  stood  in 
front  of  him  and  immediately  chases  the  former  pursuer. 
If  a  child  is  caught,  he  must  become  the  pursuer. 

DROP  THE  HANDKERCHIEF 

The  children  form  a  ring.  A  number  is  placed  in  the 
center  of  the  ring.  One  child  runs  around  the  outside 
of  the  ring  with  a  handkerchief.  He  throws  it  behind 
some  one  in  the  ring  and  calls  some  number  less  than  the 
one  in  the  ring.  The  child  behind  whom  it  is  thrown 
quickly  subtracts  the  number  called  from  the  number  in 
the  ring,  and  starts  around  the  ring  in  the  same  direction. 
If  he  overtakes  the  one  who  dropped  the  handkerchief 
before  he  reaches  the  vacant  place,  he  then  drops  the 
handkerchief.  If  he  does  not,  he  is  out  of  the  game. 

HANDS  UP 

The  children  form  a  ring  around  the  room  or  on  the 
playground.  A  number  is  pinned  on  each  child.  A 
catcher  moves  around  inside  of  the  ring.  The  children 


THE    USE   OF   GAMES   IN   NUMBER   WORK       93 

tantalizingly  hold  up  both  hands  above  their  heads, 
tempting  the  catcher  to  tap  them  while  the  hands  are 
raised.  If  he  can  succeed  in  touching  any  of  them  while 
their  hands  are  raised,  and  at  the  same  time  can  give  the 
combination  of  his  number  and  that  of  the  one  caught, 
he  takes  the  place  of  the  one  caught,  who  then  becomes 
catcher. 

HOP  AND  SHAKE  HANDS 

A  ring  is  formed  and  the  children  have  numbers  pinned 
on  them.  One  person  is  "it."  He  runs  around  the 
ring,  taps  some  one  on  the  back,  and  calls  some  number. 
Before  the  one  tapped  can  run,  he  must  add  his  number 
to  the  number  called  and  give  the  sum.  After  he  gives 
the  sum,  he  runs  in  the  opposite  direction  to  meet  the 
tapper.  When  they  meet,  they  shake  hands  and  then, 
hopping  on  one  foot,  they  try  to  see  which  one  can  get 
in  the  vacant  place  first. 

WITCH'S  CIRCLE 

One  child  is  the  witch.  All  the  children  run  across 
the  line  and,  if  the -witch  catches  a  child,  she  places  him 
in  a  small  circle  containing  a  number  combination.  If 
he  can  answer  the  combination  in  the  small  circle  as  soon 
as  he  is  placed  in  it,  he  may  go  free ;  if  not,  he  must  stay 
until  the  old  witch  releases  him. 

IN  THE  POOL 

There  is  one  child  at  each  corner  of  a  square,  making 
in  all  four  players  called  "strikers."  The  other  players 


94  THE   TEACHING  OF  ARITHMETIC 

stand  in  the  middle  of  the  square  called  "the  pool." 
The  players  in  the  pool  have  numbers  pinned  on  them. 
This  game  may  be  used  in  multiplication  or  addition. 
Let  us  suppose  that  it  is  the  4's  in  multiplication.  The 
four  players  at  the  corners  throw  a  ball  from  one  to 
another.  When  they  catch  the  ball,  they  try  to  touch  one 
of  the  children  in  the  pool  with  it.  If  that  child's  num- 
ber is  6  and  he  at  once  calls  "24,"  he  may  stay  in  the 
pool.  If  he  makes  a  mistake,  it  is  the  "striker's"  place 
to  correct  him.  If  the  "striker"  does  not  do  this,  some 
one  in  the  pool  may  call  out  the  answer.  The  child  who 
calls  the  answer  may  then  take  the  "striker's"  place. 
The  object  of  the  game  is  to  see  who  can  stay  "striker" 
the  longest. 

HAVE  You  SEEN  MY  SHEEP 

The  children  form  a  circle,  one  child  outside  being  the 
shepherd.  The  shepherd  taps  some  one  on  the  back, 
saying,  "Have  you  seen  my  sheep?"  The  one  tapped 
will  reply,  "No,  how  many  pounds  did  he  weigh?"  The 
shepherd  will  reply  with  some  such  combination  as  3x5, 
or  8+9,  or  10— 4,  and  then  start  to  run  around  the  circle. 
The  one  tapped  must  give  the  answer  to  the  combination 
before  he  can  run  after  the  shepherd,  who  tries  to  get 
around  to  the  vacant  place  before  he  is  caught.  If  the 
shepherd  is  not  caught,  the  one  tapped  becomes  shepherd. 
If  the  one  tapped  cannot  answer  his  combination  cor- 
rectly, he  must  take  his  place  in  the  center  of  the  circle. 


CHAPTER  IX 
COMMON  FRACTIONS 

SINCE  a  little  objective  work  in  fractions  is  done  in 
the  lower  grades,  before  a  systematic  study  of  the  subject 
is  taken  up  in  the  fifth  and  sixth  grades,  it  seems  best  .to 
break  up  the  discussion  into  two  parts,  the  first  relating 
to  the  work  of  the  primary  grades,  and  the  second  to  the 
work  of  the  intermediate  grades. 

I.  EARLY  WORK  IN  FRACTIONS 

The  work  done  in  fractions  in  the  first  four  grades 
is  preparatory  to  the  written  work  with  symbols  in  the 
fifth  grade.  A  clear  notion  of  fractions  cannot  be  obtained 
from  the  symbols.  The  first  work  must  be  objective. 
By  the  use  of  objects,  diagrams,  etc.,  the  child  sees  a  half, 
a  fourth,  or  any  fractional  unit  as  an  individual  unit,  just 
as  a  quart,  or  a  foot.  He  adds,  subtracts,  multiplies,  or 
divides  a  number  of  these  units  just  as  he  would  a  num- 
ber of  integral  units.  Let  him  get  the  names  of  these 
fractional  units  by  having  him  notice  the  similarity  of 
sound  between  the  number  of  equal  divisions  and  the 
names  of  the  parts.  Thus,  one  of  four  equal  parts  is  a 
fourth;  one  of  five  equal  parts,  a,  fifth;  one  of  six,  a  sixth; 
etc. 

95 


96  THE    TEACHING  OF   ARITHMETIC 

First  study  the  fractions  in  related  groups  as  halves 
and  fourths,  then  halves,  fourths,  and  eighths,  etc.,  and 
learn  the  relations  among  them.  This  should  be  done 
objectively  so  that  a  child  sees  that  a  half  equals  two 
fourths,  just  as  he  sees  that  a  quart  equals  two 
pints,  etc. 

If  the  work  is  objectively  presented,  the  child  is  able 
to  use  intelligently  the  common  fractions  before  know- 
ing the  notation  for  them.  Thus,  he  knows  that  a  pint 
is  half  of  a  quart,  that  a  quart  is  a  fourth  of  a  gallon,  etc. 
When  a  fractional  unit  is  studied,  it  should  not  only  be 
introduced  by  some  real  need  of  such  a  relation,  but 
followed  by  real  applications.  If  halves  and  fourths  are 
studied,  there  are  many  real  applications.  Thus,  if  milk 
is  10  cents  per  quart,  how  much  is  a  pint  worth  ?  If  cider 
is  20  cents  per  gallon,  how  much  is  a  quart  worth?  If 
walnuts  are  80  cents  a  bushel,  how  much  is  a  peck  worth  ? 
In  this  way,  through  handling  objects,  folding  paper, 
making  diagrams,  etc.,  the  pupil  comes  to  think  of  a 
fractional  unit  as  a  concrete  thing,  just  as  an  integral 
unit.  Thus  he  gets  the  first  of  the  three  steps  in  the 
proper  teaching  of  fractions,  viz. :  (1)  the  meaning,  (2)  the 
notation,  and  (3)  the  manipulation. 

THE  NOTATION  OF  A  FRACTION 

It  is  very  important  that  the  pupil  should  get  the  full 
meaning  of  the  notation  of  a  fraction,  for  all  the  manipula- 
tions depend  upon  the  notation.  It  is  not  so  important 
for  him -to  get  the  names  "numerator"  and  "denominator" 
when  the  notation  is  first  presented;  in  fact,  it  may  be 


'COMMON   FRACTIONS  97 

better  to  avoid  using  these  terms  until  after  the  function 
of  the  terms  is  well  understood. 

In  developing  the  notation,  take  some  object  or  dia- 
gram, and  divide  it  into  a  number  of  equal  parts,  say  4. 
Then  have  the  children  point  out  some  number  of  these 
fourths,  as  3.  Now  write  3  on  the  board  and  ask  if  that 
alone  tells  us  that  there  are  three  fourths.  Show  3  of 
some  other  unit  as  dollars,  cents,  feet,  etc.,  and  let  the 
pupils  see  that  the  3  shows  "how  many"  and  that  some 
other  sign  is  needed  to  show  what  they  are,  as  $3,  3^,  3  ft., 
etc.  Then  show  that  a  sign  is  needed  to  show  what  the 
3  fourths  are,  and  that  the  sign  is  a  4  written  below  the 
3.  Thus  in  £  the  3  shows  "how  many"  and  the  4  shows 
what  they  are. 

To  test  the  pupils'  understanding  of  the  notation,  write 
several  fractions  and  ask  of  each  fraction,  "How  many 
things?"  "What  are  they?"  as : 

8527537        3       „+ „ 

?>  ¥>  ~s>  ~s>  r,  -5,  -$>  TG,  etc. 

This  notation  is  the  basis  of  all  the  processes  and  must 
be  thoroughly  understood.  But  when  it  is  understood, 
the  manipulations  follow  from  those  of  integers. 

THE  FUNDAMENTAL  PROCESSES  IN  FRACTIONS 

In  teaching  the  processes  they  should  be  related  to 
those  already  known.  Thus,  just  as  $3-f-$l=$4,  so 
f +£=!••  Just  as  3  ft. +8  in.  cannot  be  added  until 
changed  to  a  "like  unit,"  so  |+i  cannot  be  added  until 
changed  to  a  like  unit.  In  the  same  way,  subtraction 
follows  that  of  integers. 


98  THE   TEACHING  OF   ARITHMETIC 

Just  as  3X$5  means  $5+$5+$5,  or  $15,  so  3Xf  = 
f +f +f  =  •¥•;  or  3  times  5  of  any  unit  is  15  of  that  unit. 

When  we  speak  of  multiplying  by  a  fraction,  we  are 
giving  a  new  meaning  to  multiplication.  The  process  is 
based  upon  partition,  not  upon  addition,  as  was  the  case 
when  multiplying  by  an  integer.  But  just  as  f  of  an 
object  is  found  by  first  dividing  the  object  into  4  equal 
parts,  then  taking  3  of  them;  or  f  of  a  group,  as  16, 
is  found  by  dividing  16  into  4  equal  groups,  then  taking 
3  of  the  groups ;  so  to  find  f  of  a  fraction  means  the  same 
thing.X  Thus,  f  of  f  "means  that  we  first  divide  each  of 
the  three  fifths  into  4  equal  parts,  making  three  twentieths, 
then  take  3  of  them,  making  ^.  .  That  is,  f  of  f  =  \£y. 

Also,  just  as  $8-i-$2  =  4  means  that  $8  will  contain 
$2  four  times,  so  f-j-f  =  4  means  that  f  will  contain  f 
four  times.  Likewise,  8  of  any  unit  will  contain  2  of  that 
unit  four  times. 

And,  just  as  one  cannot  tell  how  many  times  2  ft.  will 
contain  8  in.  without  first  knowing  how  many  inches  in 
2  ft.,  so  one  cannot  tell  how  many  times  one  fraction  will 
contain  another  until  they  are  both  expressed  in  the  same 
unit.  Thus,  the  first  work  in  the  division  of  fractions  is 
based  upon  the  measurement  phase  of  division  of  integers. 

H.   FRACTIONS   COMPLETED 

OBJECTIVE  PRESENTATION  OF  FRACTIONS 

It  was  shown  in  the  first  part  of  this  chapter  that  frac- 
tions are  studied  objectively  in  the  lower  grades  and  that 
pupils  gain  naturally  and  informally  one  of  the  several 


COMMON  FRACTIONS  99 

ideas  involved  in  fractions.  This  is,  that  a  fraction  is  one 
or  more  of  the  equal  parts  into  which  some  whole  has  been 
divided.  This  notion  is  the  basis  upon  which  the  funda- 
mental processes  are  built. 

In  the  lower  grades  and  through  the  use  of  objects,  the 
child  also  begins  to  get  one  of  the  real  uses  of  fractions, 
viz.,  its  use  as  a  means  of  expressing  a  relation.  He 
knows  that,  if  he  has  4  marbles  and  loses  one  half  of  them, 
he  will  have  but  2  left;  and  also  that,  if  he  has  4 
marbles  and  loses  2  of  them,  he  has  lost  half  of  them. 
This  use  of  a  fraction  to  express  a  relation  comes  slowly 
and  gradually  to  him  through  such  related  questions 
as,  "If  you  have  6^  and  spend  half  of  them,  how  much 
will  you  spend?"  and,  "If  you  have  6£  and  spend  3j£, 
what  part  of  your  money  do  you  spend?" 

* 
THE  IDEAS  INVOLVED  IN  A  FRACTION 

There  are  three  ideas  involved  in  a  fraction :  (1)  a 
fraction  is  one  or  more  of  the  equal  parts  of  a  whole ;  (2) 
a  fraction  is  an  indicated  division;  and  (3)  a  fraction  is 
an  expression  of  the  ratio  of  one  quantity  to  another. 
The  development  of  the  fundamental  processes  from  the 
corresponding  processes  with  whole  numbers  depends 
upon  the  first  of  these  ideas.  Much  of  the  pupil's  later 
work  depends  upon  the  second  and  third  ideas,  and  these 
must  be  firmly  fixed  in  his  mind  before  the  completion 
of  the  subject ;  but  they  need  not  be  taken  up  until  after 
the  development  of  the  fundamental  processes  with  frac- 
tions —  addition,  subtraction,  multiplication,  and  division. 


100  THE    TEACHING  OF   ARITHMETIC  \ 

A  FRACTION  AS  A  PART  OF  A  WHOLE 

First  through  dividing  objects,  folding  paper,  using 
diagrams  upon  the  blackboard,  or  with  similar  illustra- 
tions, get  the  pupil  to  see  the  principle  involved  in  naming 
the  fractional  unit.  That  is,  have  him  see  that,  if  a  thing 
is  divided  into  four  equal  parts,  each  part  is  a  fourth  of 
the  whole;  if  in  fine  equal  parts,  each  is  &  fifth;  if  in  six, 
each  is  a  sixth;  etc.  Next  have  him  count  the  fourths, 
fifths,  sixths,  etc.,  just  as  he  would  count  any  unit,  and 
thus  f,  $,  $-,  etc.,  will  become  just  as  real  numbers  of  things 
as  3  feet,  4  quarts,  and  5  bushels  are. 

It  is  now  important  that  the  pupil  get  the  correct 
idea  of  the  notation  from  this  point  of  view  of  a  fraction 
as  a  part  of  a  whole.  When  he  points  out  a  number  of 
fractional  units  as  3  or  4  of  them,  have  him  see  that  just 
as  in  three  dollars  or  three  feet,  3  only  tells  how  many, 
and  that,  to  express  the  whole  idea,  the  sign  of  the  unit 
is  necessary  as  $3f  and  3  ft. ;  so  in  fractions  three  fourths 
is  written  f,  the  3  denoting  the  number  of  things  and  the 
4  telling  what  they  are,  just  as  "$"  or  "ft."  does. 

A  little  drill  upon  such  fractions  as  £,  •£,  f,  f,  -fa,  etc., 
asking,  "How  many?"  pointing  to  the  numerator;  then, 
"What  are  they?"  pointing  to  the  denominator,  will 
soon  fix  in  mind  the  function  of  the  two  terms  of  a  frac- 
tion, a  thing  much  more  important  than  to  know  their 
names. 

Knowing  the  meaning  of  the  terms,  however,  the  names 
may  be  so  given  as  to  make  it  easy  to  remember  them. 
Thus,  writing  some  fractions  on  the  blackboard,  the 


COMMON   FRACTIONS  101 

pupil  will  tell  you  that  the  upper  term  tells  "how  many" 
or  the  number  of  things.  It  is  then  the  numberer  or  numer- 
ator, and  through  the  similarity  in  sound  he  fixes  the 
name  and  its  meaning.  He  then  tells  you  the  lower 
term  tells  what  the  number  of  things  denoted  by  the 
upper  term  are.  Get  him  to  see  then  that  it  names  the 
unit  or  is  the  namer  and  hence  is  called  the  denominator. 
The  pupil  can  remember  this  if  the  teacher  will  call  his 
attention  to  the  fact  that  when  we  name  a  man  to  run  for 
office  we  nominate  him,  and  that  we  are  "  the  nominators  " ; 
and  likewise  the  term  that  names  the  fraction  is  the 
de-nominator. 

A  FRACTION  AS  AN  INDICATED  DIVISION 

It  is  essential  in  changing  common  fractions  to  deci- 
mals, and  in  other  work  that  follows,  that  a  child  get  a 
fixed  impression  of  a  fraction  as  an  indicated  division^ 
but  it  is  hard  to  give  him  a  satisfying  basis  for  this  notion. 
It  is  as  well  to  make  no  attempt  to  give  such  an  idea  until 
after  the  processes  have  been  mastered.  But,  having 
mastered  the  processes,  he  knows  that  finding  %  of  any- 
thing means  to  divide  it  into  4  parts.  So  i  of  8  means 
8-:- 4.  He  also  knows  that  £  of  1  is  -j,  and  that  £  of  8 
is  eight  times  as  large,  and  hence  is  f .  From  this  and 
similar  examples  get  him  to  see  the  truth  of  the  following 
statements : 

iof  8=  |,  and  =  8-5-4; 
•^  of  6=  f ,  and  =  6^3; 
£  of  15=^,  and  =15-5-5. 


102  THE   TEACHING  OF  ARITHMETIC 

Hence, 

f=  84-4; 

*-  6-4-3; 

^=15-J-5. 

Follow  this  by  having  the  pupils  express  such  forms  as 
3-5-4,  5-f-7,  3-7-8,  2-r-5,  etc.,  as  fractions. 

A  FRACTION  AS  A  RATIO 

This  conception  is  a  slow  growth  that  comes  from  using 
a  fraction  to  express  the  relations  of  simple  objects  from 
the  very  first  conception  of  a  fraction.  The  pupil  may 
get  much  help,  however,  from  the  meaning  of  the  flota- 
tion of  a  fraction.  Thus,  £  is  5  of  the  8  equal  parts; 
that  is,  it  is  5  of  8.  So,  if  the  pupil  has  eaten  5  of  8  apples, 
he  has  eaten  "5  of  8"  or  f  of  them.  If  he  has  11  marbles 
and  loses  8,  he  has  left  3  of  11  or  -n>  And  thus  he  comes 
gradually  to  see  that  the  ratio  of  one  number  to  another 
is  the  fraction  whose  numerator  is  the  first  number  and 
whose  denominator  is  the  second. 

Objectively  the  pupil  may  see  that  the  ratio  idea  is 
consistent  with  his  first  idea  of  a  fraction  as  one  or  more  of 
the  equal  parts  of  a  whole. 

I 


Thus,  if  we  consider  9  things,  say,  as  the  whole,  then  each 
is  one  of  the  nine  equal  parts  that  made  the  whole,  or 
£  of  the  whole.  And  7  of  them  is  £  of  the  whole,  and  thus 
the  ratio  of  7  to  9  is  $ ;  that  is,  7  is  %  of  9. 


COMMON   FRACTIONS  103 

ADDITION  OF  FRACTIONS 

After  the  nature  and  notation  of  a  fraction  is  under- 
stood, the  addition  of  fractions  presents  nothing  new; 
that  is,  the  work  has  the  same  meaning  and  follows  the 
same  fundamental  law,  that  only  like  things  can  be  added, 
which  the  pupil  has  had  in  whole  numbers.  The  impor- 
tant thing  is  that  the  work  be  carefully  graded  so  as  to 
introduce  but  one  form  at  a  tune  and  that  the  pupil 
sees  that  it  is  like  the  addition  he  already  knows.  The 
gradation  should  be :  (1)  two  fractions  whose  units  are 
alike ;  (2)  two  fractions  whose  units  are  unlike,  but  where 
one  can  be  changed  to  the  other ;  (3)  two  fractions  whose 
units  are  unlike  and  a  new  unit  must  be  found  to  which 
both  can  be  changed.  ' 

ADDING  FRACTIONS  WITH  LIKE  UNITS 

The  pupil  knows  that  the  upper  term,  or  numerator, 
shows  the  number  of  things  considered  and  that  the  lower 
term,  or  denominator,  shows  what  they  are.  So  just 
as  $2+$3  =  $5,  or  2  ft. +3  ft.  =  5  ft.,  so  2  of  any  unit  +3 
of  a  like  unit  =5  of  that  unit.  Hence,  f+f=f,  or  f+f 
=  f,  etc.  And  thus  he  should  see  clearly  that  when  the 
units  are  alike,  the  numerators  only  are  added  for  they 
are  the  number  of  things  that  are  combined.  The  de- 
nominator merely  shows  what  they  are  as  do  the  abbre- 
viations and  signs  in  denominate  numbers. 

If  the  work  is  presented  in  this  way,  pupils  will  never 
attempt  to  add  two  fractions  by  adding  numerators  and 
denominators,  a  mistake  often  seen  in  grammar  and  high 
schools. 


104 


THE   TEACHING  OF   ARITHMETIC 


ADDING  FRACTIONS  WITH  RELATED  DENOMINATORS 

The  pupil  knows  from  his  work  in  whole  numbers  and 
with  denominate  whole  numbers  that  2  bu.+3  pk.  does 
not  equal  5  bu.  or  5  pk. ;  that,  before  he  can  combine 
them  into  one  number,  they  must  both  be  expressed  in 
like  units,  and  that  he  must  express  them  as  8  pk.+3  pk. 
before  he  can  add  them.  So  in  i+f,  he  knows  that, 
although  he  has  1  thing  +  3  things,  he  cannot  call  them 
4  of  either,  for  1  of  them  is  a  fourth  while  the  other  3  are 
eighths.  So,  before  they  can  be  added,  the  i  must  be 
expressed  as  f.  Then  f+ !=•§•• 

It  will  be  noticed  that  this  class  of  addition  calls  for  a 
knowledge  of  the  relations  of  certain  related  fractional 
units.  While  a  study  of  the  idea  of  a  fraction  as  one  or 
more  of  the  equal  parts  of  a  whole  through  the  use  of  ob- 
jects has  no  doubt  shown  the  pupil  certain  relations,  this 
class  of  addition  furnishes  a  motive  for  further  comparison, 
and  the  pupil  should  now  become  as  familiar  with  the 
relations  of  those  units  likely  to  occur  in  life  as  he  is  with 
the  relation  of  the  units  in  the  various  tables  of  denominate 
numbers.  This  he  does  through  the  use  of  objects  and 
diagrams. 

Thus,  a  diagram  like  the  following  shows  him  the  rela- 
tion of  halves,  fourths,  and  eighths. 


a 

y, 

X 

1A 

l/4 

% 

% 

y* 

£ 

H 

ys 

% 

% 

% 

COMMON   FRACTIONS 


105 


The  following  shows  the  relation  of  thirds,  sixths,  and 
ninths. 


>/3 

>/3 

14 

% 

*/6 

% 

IA 

H 

Vo 

% 

% 

y9 

% 

*/9 

'A 

% 

*/9 

% 

ADDING  FRACTIONS  WITH  UNRELATED  UNITS 

The  adding  of  such  fractions  asf  and^  brings  up  the 
auxiliary  process  of  changing  fractions  to  like  units  or  to 
"a  common  denominator."  Such  a  problem  as  this, 
then,  should  be  presented  in  order  to  furnish  an  ami  or 
motive  for  such  a  process. 


THE  LEAST  COMMON  DENOMINATOR 

Using  the  problem  presented;    namely,  f+f,  let  us 
find  a  unit  to  which  both  3ds  and  4ths  can  be  changed. 


By  diagrams  show  that  if  3ds  are  divided  into  2  equal 
parts  we  have  6ths;  if  into  3,  we  have  9ths;  if  into  4, 
we  have  12ths ;  etc.  Have  the  pupil  see  that  the  denomi- 
nators of  the  fractions  to  which  3ds  can  be  changed  are 
the  successive  multiples  of  3  and  thus  introduce  the  term 


106  THE   TEACHING   OF   ARITHMETIC 

multiple.  If  that  is  always  so,  then  4ths  can  be  changed 
to  8ths,  12ths,  16ths,  20ths,  etc.  Let  him  see  from  the 
diagram  that  that  is  just  what  he  would  have. 

He  is  now  led  to  see  that  the  denominators  to  which 
3ds  and  4ths  can  be  changed  will  contain  both  3  and  4. 
He  should  now  be  able  to  find  by  inspection  the  unit 
to  which  two  or  more  fractional  units  can  be  changed. 

REDUCING  FRACTIONS  TO  NEW  UNITS 

Using  the  diagrams  above,  lead  the  pupil  to  see  that 
when  3ds  are  changed  to  6ths,  while  the  units  are  half 
as  large,  there  are  twice  as  many  of  them;  changed  to 
9ths,  they  are  one  third  as  large  but  three  times  as  many ; 
etc.  Thus,  f  =  $  =  £  =  •&  =  T$,  etc.,  which  may  be  found 
mechanically  by  multiplying  both  terms  by  the  same 
number. 

Here  it  is  well  to  call  attention  to  the  use  he  will  have 
of  the  truth  illustrated  here  —  that  both  terms  of  a  frac- 
tion may  be  multiplied  or  divided  by  the  same  number 
without  changing  the  value  represented  by  the  fraction. 

Now,  to  change  f  and  f  to  like  units,  ask,  "What 
denominator  will  contain  both  3  and  4  ?  "  "  By  what  must 
3  be  multiplied  to  give  12?"  "Then  by  what  must 
each  term  of  f  be  multiplied  to  change  it  to  12ths?" 


and  then  ask  similar  questions  about  f. 

REDUCING  IMPROPER  FRACTIONS 
When  adding  the  fractions  proposed  above,  we  have 


^.     Now   the  pupil  will    observe    that 
is  not  a  fraction  in  the  sense  that  he  has  thought  of  a 


COMMON   FRACTIONS  107 

fraction,  for  in  dividing  a  whole  into  12  parts  to  get  12ths 
there  are  but  12  of  them  in  any  whole.  So  this  may  be 
called  an  improper  fraction,  since  12  of  the  17  twelfths 
will  make  a  whole  and  5  of  them  will  remain.  Hence, 
T!  =!•&•  Unless  the  pupil  has  come  to  the  second  no- 
tion of  a  fraction,  that  it  is  an  expressed  division,  he  sees 
no  meaning  in  "  dividing  the  numerator  by  the  denomi- 
nator" as  a  method  of  reduction.  The  natural  develop- 
ment from  his  first  idea  of  a  fraction  is  that,  since  the 
denominator  shows  the  number  of  parts  into  which  the 
whole  was  divided,  it  is  the  number  of  parts  that  it  takes 
to  make  a  whole.  Thus,  in  ^,  it  takes  8  of  the  8ths  to 
make  a  whole ;  so  there  will  be  as  many  wholes  as  there 
are  8's  in  25,  or  3  of  them  and  1  remaining.  So  ^  =  3£. 
Through  such  a  thought  as  this,  he  sees  that  mechani- 
cally the  answer  is  obtained  by  "dividing  the  numerator 
by  the  denominator,"  the  rule  usually  given. 

FORMS  OF  ADDING  MIXED  NUMBERS 

There  are  several  forms  of  adding  mixed  numbers. 
The  two  most  common  are  given  below : 

Form  A  Form  B 

24 


_  10    3 


16|=16H 


17* 


3 
18 
20 


41 

Form  A  is  an  uneconomical  procedure  for  it  requires 
rewriting  the  whole  numbers  and  the  common  denominator 


108  THE   TEACHING  OF  ARITHMETIC 

with  each  fraction.  In  form  B  the  common  denominator 
is  written  above  and  the  numerators  are  written  in  a  form 
easily  added. 

ADDING  SPECIAL  FRACTIONS 

The  pupils  should  always  be  alert  for  special  combi- 
nations that  will  save  work.  There  are  two  types  of  exer- 
cises in  the  addition  of  fractions  that  should  be  noticed. 
These  two  types  are  shown  by  the  following  examples : 

First  type:  £+t+|+f+i 

Instead  of  changing  all  to  8ths,  the  pupil  should  collect 
those  whose  units  are  alike.  So  f+i  =  l;  f+t=l; 
and,  hence,  the  sum  =2£. 

Second  type:  i+i. 

Since  4  and  5  have  no  common  factor  and  since  the  nu- 
merators are  each  1,  the  new  numerators  will  be  5  and  4 
respectively.  Hence,  the  sum  is  -fo  which  is  "  the  sum  of 
the  denominators  over  their  product." 

SUBTRACTION  OF  FRACTIONS 

As  in  addition  of  fractions,  the  pupil  should  see  that 
the  meaning  and  the  process  of  subtraction  are  the  same  as 
in  whole  numbers.  The  gradation  is  not  so  important  as 
in  addition,  for  all  the  principles  needed  have  already 
been  taught.  Yet,  the  grading  is  exactly  the  same  as  in 
addition  except  that  the  passing  from  one  type  to  another 
may  be  done  more  quickly.  The  grading  is : 

(a)  Like  units,  asf— f  =  f=i; 

(6)  Related  units,  as  f — i  =  f—  f  =  f; 

(c)  Unrelated  units,  as  f — i  =  •& — TS  =  &. 


COMMON  FRACTIONS  109 

FORMS  OF  WORK  IN  MIXED  NUMBERS 

When  the  fraction  in  the  minuend  is  greater  than  the 
one  in  the  subtrahend,  they  are  subtracted  as  in  pure 
fractions.  With  pupils  in  the  lower  grades  the  work  may 
be  written  down  as  in  form  B  under  addition.  With 
mature  pupils  the  change  to  like  units  and  the  subtrac- 
tion may  be  done  mentally  and  the  numerators  of  the 
like  fractions  not  written.  That  is, 

46|  46} 


13!       .      not 


8 
3 

5 

When  the  fraction  in  the  minuend  is  smaller  than  the 
one  in  the  subtrahend,  work  is  saved  if  the  fraction  in 
the  subtrahend  is  subtracted  from  1  and  the  result  added 
to  the  fraction  of  the  minuend.  Thus,  in 

we  think 


Now  we  may  use  1—  f  +i  =  !+i=:Tj,  instead  of  1-J—  f  = 


SUBTRACTING  SPECIAL  FRACTIONS 

When  the  numerators  are  each  1  and  the  denominators 
have  no  common  factor,  the  result  may  be  written  down 
at  once.  Thus  !—  !  =  i%,  for  the  new  numerators  are  5 
and  2  respectively  and  the  common  denominator  is  10; 
that  is,  the  result  is  "the  difference  of  the  denominators 
over  their  product." 


110 


THE   TEACHING  OF  ARITHMETIC 


MULTIPLICATION   OF  FRACTIONS , 

MULTIPLYING  A  FRACTION  BY  A  WHOLE  NUMBER 

The  pupil  should  be  shown  clearly  that  multiplying 
by  a  whole  number  and  multiplying  by  a  fraction  have 
very  different  meanings.  He  has  learned  that  multiply- 
ing by  a  whole  number  is  a  means  of  saving  addition' when 
the  addends  are  alike.  So,  just  as  3X$2  =  $6,  3X2  ft.= 
6  ft.,  and  3X2  of  any  unit  =6  of  that  unit,  so  3Xf  =  $, 
3 X •§•  =  $,  etc.;  that  is,  the  numerator  being  the  number 
of  units  under  consideration  is  the  number  that  is  multi- 
plied. The  denominator  is  merely  written  to  show  what 
the  units  are,  as  were  the  "  $  "  and  "  ft."  in  the  examples 
shown  above. 

MULTIPLYING  A  FRACTION  BY  A  FRACTION 

To  find  the  product  of  fX|  means  to  find  f  of  $.  But 
to  find  f  of  anything  always  means  the  same.  It  means 


*/ 

A 


to  divide  the  thing  into  3  equal  parts  and  then  take  2  of 
them.     Now,  5ths  divided  into  3  equal  parts  gives  15ths, 


COMMON  FRACTIONS  111 

so  i  of  £  =  A-  Hence,  f  of  f=2  X  A  =  &•  Hence,  it 
is  seen  that  the  product  of  two  fractions  is  found  by  tak- 
ing the  product  of  the  numerators  for  the  numerator  of 
the  product,  and  the  product  of  the  denominators  for  the 
denominator  of  the  product.  The  above  product  may  be 
shown  objectively. 

MULTIPLYING  BY  A  MIXED  NUMBER 

To  multiply  by  a  mixed  number  will  involve  multiply- 
ing a  whole  number  by  a  fraction.  As  seen  above,  to 
multiply  anything  by  a  fraction  requires  both  a  division 
and  a  multiplication.  But,  just  as  4^2  X3  gives  the  same 
result  as  4X3-:-  2  (that  is,  the  order  may  be  changed), 
the  multiplication  by  the  numerator  may  take  place 
before  the  division.  Thus,  13f  X25  is  found  as  in  the 
margin. 

Enough  drill  should  be  given  to  fix  the  habit         25 
of   putting   the   partial    products   in   the  right          13f 
place.     Some  pupils  will  put  the  5  of  the  second       4)75 
product  under  the  1  of  the  first.  18f 

Where  both  multiplier  and  multiplicand  con-          75 
tain  fractions  and  the  whole  numbers  are  small,          25 
work  may  be  saved  by  changing  both  to  improper        343f 
fractions.    Thus,   3iXl2f  = 


The  pupil  who  has  understood  the  two  meanings  of 
division  of  whole  numbers  will  have  no  difficulty  in  see- 
ing the  corresponding  meanings  when  applied  to  fractions, 
which  to  him  are  now  a  sort  of  denominate  number. 


112 


THE   TEACHING  OF  ARITHMETIC 


DIVIDING  A  FRACTION  BY  A  WHOLE  NUMBER 

Any  division  by  an  abstract  whole  number  may  be  inter- 
preted4i8  the  partition  idea  of  division;  that  is,  as  a  divi- 
sion into  parts.  Just  as  $8  -j-2  =  $4,  8  ft.  ^2=  4  ft,,  and 
8  of  any  unit  -i-2=4  of  that  unit,  so  f-7-2  =  $,  f-j-2=^, 
etc. 

*/ 

/4 


Likewise,  from  the  same  meaning  of  dividing  into 
parts,  each  fractional  unit  may  be  divided  into  parts 
as  shown  above.  So,  when  the  numerator  cannot  be 
divided  without  a  remainder,  the  denominator  may  be 
multiplied  by  the  divisor. 

This  last  fact  is  also  shown  from  the  same  meaning; 
that  is,  to  divide  by  2  is  to  find  half  of;  by  3,  to  find  a 
third  of;  by  4,  to  find  a  fourth  of;  etc.  Thus,  |-r-5  = 


DIVIDING  A  FRACTION  BY  A  FRACTION 

To  divide  by  a  fraction  cannot  mean  partition,  for  a 
thing  cannot  be  divided  in  f  equal  parts.  Such  an  ex* 
pression  has  no  meaning.  A  thing  can  only  be  divided 


COMMON   FRACTIONS  113 

into  2  or  3  or  4  or  some  whole  number  of  equal  parts. 
Hence,  this  type  of  division  is  the  measuring  idea  of  divi- 
sion; that  is,  it  is  finding  how  many  times  the  dividend 
will  contain  the  divisor.  Just  as  8  qt.-?-2  qt.  =  4,  or 
8  ft.  -5-  2  ft.  =4,  so  8  of  any  unit  will  contain  2  of  that 
unit  4  times.  Then,  f-r-f=4,  f-r-f  =  4,  etc.;  that  is, 
just  as  in  any  denominate  number,  the  units  must  be  alike, 
then  the  numbers  of  them  are  the  numbers  used  in  the 
actual  division. 


DIVIDING  BY  INVERTING  THE  DIVISOR 

There  are  several  ways  of  developing  the  in  verted  di- 
visor method  of  division.  Perhaps  the  simplest  method 
is  to  analyze  the  work  done  in  changing  to  like  units. 
Thus,  take  the  example  given  above  of  f-5-f.  To  get 
like  units,  both  terms  of  the  dividend  were  multiplied  by 
the  denominator  of  the  divisor.  Then  both  terms  of  the 
divisor  were  multiplied  by  the  denominator  of  the  divi- 
dend. But  when  thus  reduced,  only  the  new  numerators, 
2X4-4-3X3,  were  actually  used  in  the  division.  This 

2X4 

quotient  expressed  as  a  fraction  is  -  ,  which  is  the 

0X0 

product  of  f  Xi-    But  this  is  the  product  of  the  dividend 
by  the  divisor  inverted. 

Another  method  in  quite  common  use  is  to  develop  the 
process  from  the  two  known  facts  that  multiplying  the 
divisor  divides  the  quotient,  and  dividing  the  divisor 


114  THE   TEACHING  OF   ARITHMETIC 

multiplies  the  quotient;    and  that  with  the  divisor  re-. 
maining  constant,  multiplying  or  dividing  the  dividend 
multiplies  or  divides  the  quotient  by  the  same  number. 
With  these  two  principles  hi  mind,  the  development  is  as 
follows  : 

i+i-ij 

1*1=4X1=4; 


From  several  examples  like  this,  the  pupil  gets  the  follow- 
ing generalization,  that  the  quotient  of  one  divided  by  any 
fraction  is  the  fraction  inverted.  The  second  step  is  as 
follows  : 

Since     I-^T=& 

Then    i+f-fX*  =  t. 

That  is,  to*  divide  by  a  fraction,  invert  the  divisor  and 
multiply. 

DIVIDING  A  MIXED  NUMBER  BY  A  WHOLE  NUMBER 

While  no  new  principles  are  involved,  the  method  of 
dividing  a  mixed  number  by  a  whole  number  needs  to  be 
taken  up.  Thus,  to  divide  3896f  by  7,  we  have 

7)3896f 

556,  4f  rem. 

4f^-7  =  |XY=f-    Hence,  quotient  =  556| 

That  is,  the  division  is  performed  as  in  whole  numbers 
until  a  remainder  less  than  the  divisor  is  found  and  then 
this  is  treated  as  any  fraction  divided  by  a  whole  number. 


COMMON   FRACTIONS  115 

DIVIDING  A  MIXED  NUMBER  BY  A  MIXED  NUMBER 

There  are  two  ways  of  proceeding.  When  the  whole 
numbers  are  small,  the  mixed  numbers  may  be  changed 
to  improper  fractions.  When  the  whole  numbers  are 
large  and  the  terms  of  the  fractions  small,  time  is  saved 
by  multiplying  both  dividend  and  divisor  by  the  least 
common  multiple  of  the  denominators  and  thus  reducing 
them  to  whole  numbers.  Thus, 

21    .    03  5  S/_§_  —  25  . 
TS~  <%  —  ?XTS  —  ^7» 

And    345i-r-16f 
4       4 


Or,  a  third  method  may  be  used  in  such  a  problem  as 
468Tf-*-3-|.  Multiplying  both  dividend  and  divisor  by 
2,  we  have  937^  -=-7. 

7)937jf 
133, 


COMPLEX  FRACTIONS 

A  complex  fraction  is  merely  an  indicated  division, 
expressed  in  the  form  of  a  fraction  instead  of  by  the  di- 
vision sign  (-i-),  where  one  or  both  of  the  terms  (dividend 
and  divisor)  are  fractions  or  mixed  numbers.  Such  ex- 
pressions arise  in  expressing  the  work  to  be  done  in  the 
solution  of  a  problem  or  in  dealing  with  a  remainder  as 
shown  in  some  of  the  above  divisions.  Such  fractions, 
then,  involve  nothing  new  and  are  simplified  by  perform- 


116  THE   TEACHING  OF  ARITHMETIC 

ing  the  indicated  divisions,  or  by  multiplying  both  terms 
of  the  complex  fraction  by  the  least  common  denominator 
of  the  fractions  composing  the  terms.  Thus, 

6J=  13     J5_=39 
7*      2      22      44 


7*     7iX6      44 


Among  the  topics  often  urged  for  elimination  is  the 
topic  of  "complex  fractions,"  but  there  seems  to  be  no 
more  reason  for  this,  when  a  complex  fraction  is  considered 
as  a  mere  form  of  expressing  division,  than  to  urge  that 
no  division  must  ever  be  expressed  as  a  fraction.  The 
thing  to  be  urged  is  that,  as  in  all  work,  no  more  complex 
exercises  be  given  for  pure  computation  than  may  arise 
in  solving  the  problems  that  arise  in  doing  the  ordinary 
life  work. 


CHAPTER  X 


OURS  is  a  decimal-place-value  system  of  notation.  Ten 
units  of  any  order  make  one  Unit  of  the  next  higher  order. 
Or,  expressing  the  same  thing  in  another  way,  a  unit  in  any 
order  has  one  tenth  of  the  value  of  a  unit  in  the  next  higher 
order.  The  notation,  then,  of  a  decimal  fraction  should  be 
considered  as  an  extension  of  our  notation  to  the  right  of 
ones'  place.  Then  the  first  order  to  the  right  of  ones'  place 
is  tenths;  the  next  is  tenths  of  tenths  or  hundredths ;  the 
next,  tenths  of  hundredths  or  thousandths;  etc. 

This  notation  should  be  presented  by  taking  some 
number  all  of  whose  digits  are  alike  and  asking  the  pupil 
the  value  represented  by  each  digit  and  by  having  him 
compare  their  values.  Thus,  in  $1111,  ask  what  each  of 
the  four  1's  represents.  Next  compare  one  with  another 
and  bring  out  the  fact  that  each  represents  one  tenth  of 
the  value  of  the  one  to  the  left  of  it.  Now,  calling  any 
digit  a  certain  unit,  get  the  pupil  to  see  that  he  then  knows 
what  each  unit  must  be,  since  each  has  one  tenth  the  value 
of  the  one  to  the  left  of  it. 

Next  take  some  number  as  3467  and  have  some  pupil 
read  it.  Now  suppose  that  the  4  is  in  ones'  place;  get 
the  pupil  to  see  that  all  the  other  places  are  known  and 
that  3  must  then  be  in  tens'  place,  that  the  unit  of  the 

117 


118  THE   TEACHING  OF   ARITHMETIC 

place  occupied  by  6  must  be  one  tenth  of  one  or  tenths, 
and  that  7  must  be  in  hundredths'  place,  for  a  tenth  of  a 
tenth  is  a  hundredth.  And  thus  introduce  the  use  of 
the  decimal  point  to  locate  ones'  place. 

Then  practice  in  reading  and  writing  decimals,  and 
mixed  decimals  should  follow  until  the  pupil  visualizes 
the  position  of  each  figure  as  a  number  is  read. 

Have  pupils  see  clearly  that  each  zero  written  be- 
tween a  decimal  point  and  a  digit  that  follows  divides 
the  number  by  10  for  it  moves  the  digit  to  a  lower  order. 
Thus  if  .5  is  changed  to  .05  or  .005,  it  is  changed  to  a 
number  ^  and  y^y  as  large,  respectively. 

But  if  a  zero  is  annexed  to  a  decimal,  it  does  not  change 
its  value,  for  it  does  not  change  the  order  of  any  of  the 
digits.  Thus,  .5  may  be  changed  to  .500,  which  is  still 
5  tenths  and  no  hundredths  and  no  thousandths,  without 
changing  its  value. 

ADDITION  AND  SUBTRACTION  OF  DECIMALS 

The  pupil  who  understands  the  fundamental  principle 
of  addition  and  subtraction  —  that  only  like  units  can 
be  added  or  subtracted  —  and  who  understands  the 
notation  of  a  decimal  will  see  that  the  decimal  points 
must  come  under  each  other  in  order  to  bring  like  units 
in  the  same  columns.  Hence,  these  two  processes  present 
nothing  new  and  introduce  no  difficulties. 

MULTIPLICATION  OF  DECIMALS 

The  only  difficulty  arising  in  the  multiplication  of 
decimals  not  already  encountered  in  whole  numbers  is 


DECIMAL   FRACTIONS  119 

the  pointing  off  of  the  product.  This  is  easily  overcome 
by  proper  gradation.  The  work  should  be  rationalized, 
showing  how  it  follows  naturally  from  the  meanings  of 
multiplication  by  a  whole  number  and  by  a  fraction, 
already  discussed. 

MULTIPLYING  A  DECIMAL  BY  A  WHOLE  NUMBER 

Multiplying  by  a  whole  number  is  always  an  abridged 
way  of  finding  the  sum  of  a  number  of  equal  addends. 
Hence,  4X7.38  means  7.38+7.38+7.38+7.38  or  29.52. 
Thus,  it  is  seen  that  multiplying  a  decimal  by  a  whole 
number  must  give  a  product  with  the  same  number  of 
decimal  places  as  there  were  in  the  multiplicand. 

MULTIPLYING  A  DECIMAL  BY  A  DECIMAL 

Since  to  multiply  by  any  fraction  is  to  perform  both 
a  division  and  a  multiplication,  dividing  a  decimal  by 
10,  100,  1000,  etc.,  must  be  understood  before  developing 
this  phase  of  multiplication.  This  division  goes  back  to 
the  meaning  of  the  notation  of  a  decimal.  Thus,  5.34  is 
but  one  tenth  as  large  as  53.4,  for  each  digit  occupies  an 
order  one  lower  in  5.34  than  in  53.4.  Likewise  5.34  is  one 
hundredth  as  large  as  534.  for  each  digit  occupies  a  place 
two  orders  lower.  Thus,  it  is  seen  that  every  order  to 
the  left  to  which  the  decimal  point  is  moved  brings  each 
digit  down  to  the  next  lower  order  and  thus  divides  the 
number  by  10. 

.4X34.2  means  4  times  one  tenth  of  34.2.  This  re- 
quires moving  the  point  one  place  to  the  left  and  multiply- 
ing the  result  by  4.  But  the  multiplication  may  take 


120  THE   TEACHING  OF   ARITHMETIC 

place  first.  Hence,  by  a  few  such  illustrations,  it  is  easily 
seen  that  the  numbers  are  multiplied  as  if  they  are  whole 
numbers  and  then  as  many  decimals  pointed  off  in  the 
product  as  there  are  in  both  factors. 

DIVISION  OF  DECIMALS 

The  new  difficulty  arising  in  division  of  decimals  is  the 
pointing  off  of  the  quotient,  and  this  is  easily  developed 
by  the  proper  gradation.  The  first  problem  should  be 
the  division  by  a  whole  number,  before  division  by  a 
decimal  is  considered. 

DIVISOR  A  WHOLE  NUMBER 

Division  by  an  abstract  whole  number  may  always  be 
interpreted  as  the  partition  idea  of  division,  which  gives 
a  quotient  of  the  same  unit  as  the  dividend.  Thus,  just 
as  $8*  2  =  $4,  8  ft.H-2  =  4  ft,  f^2  =  |;  so  .8-5-2«.4, 
.08  4- 2  =  .04,  etc. 

With  this  understanding  of  the  meaning  of  division, 
the  pupil  who  has  been  taught  from  the  first  to  write 
each  digit  of  the  quotient  directly  over  the 
right-hand   digit    of    the    partial   dividend,  34J5 

from  which  it  was  obtained,  will  experience     53)1833.8 
no  new  difficulty  whatever.    Thus,  to  divide          159 
1833.8  by  53,  the  pupil  sees  that,  since  the  243 

first  dividend  was  183  tens,  the  quotient  was  212 

tens,  the  next  quotient  ones,  the  next  tenths,  31  8 

and  that  the  decimal  point  in  the  quotient  31  8 

must  come  directly  over  the  decimal  point  in 
the  dividend  in  order  to  show  this. 


DECIMAL   FRACTIONS  121 

DIVISOR  A  DECIMAL 

The  pointing  off  in  this  case  is  more  easily  developed 
by  changing  to  a  new  problem  whose  divisor  is  a  whole 
number  and  thus  make  the  pointing  depend  _  fi 

upon  the  partition  idea  as  in  the  preceding  09x1709 
case.  This  will  require,  first,  that  the  pupils  -  ' 

see  that  a  thing  could  not  be  divided  into,  say,          TQ9 
3.2  equal  parts;  and,  next,  that  multiplying  1  „„ 

both  dividend  and  divisor  by  the  same  num-  

ber  does  not  change  the  quotient.  The  first  may  be 
shown  objectively  and  the  second  by  such  illustrations 
as  6-r-2  =  3;  60*20  =  3;  600-r- 200  =  3,  etc.,  and  thus 
that  17,92*  3.2  =  179.2-*- 32.  In  the  actual  work  it  is 
not  desirable  to  erase  the  original  decimal  points.  They 
may  be  needed  in  checking  up  the  original  problem. 
But  the  new  position  in  the  dividend  may  be  marked  by 
a  caret  to  show  the  position  of  the  point  in  the  quotient. 

The  pupil  should  see  that,  when  the  dividend  has  fewer 
places  than  the  divisor,  as  many  or  more  places  may  be 
created  by  bringing  down  zeros ;  for  this  does  not  change 
the  value,  but  merely  records  the  fact  that  no  units  of 
those  orders  exist. 

CHANGING  COMMON  FRACTIONS  TO  DECIMALS 

It  is  important  that  the  pupil  be  able  to  change  a  com- 
mon fraction  or  a  ratio  to  a  decimal  fraction,  but  this 
requires  no  new  fact  or  process.  It  comes  from  consider- 
ing a  fraction  or  ratio  as  an  expressed  division  and  then 
performing  the  division. 


122  THE   TEACHING  OF   ARITHMETIC 

There  are  common  fractions  with  special  denominators 
that  are  more  easily  changed  by  first  changing  to  a  com- 
mon fraction  whose  denominator  is  a  power  of  ten.  Thus, 


THE  APPLICATION  OF  DECIMALS 

As  in  all  subjects,  the  subject  of  decimals  should  be 
used  to  meet  the  pupil's  needs  if  possible.  However, 
his  own  personal  needs  rarely  ever  require  a  knowledge 
of  decimals,  but  statistics  and  data  met  in  other  school 
work  and  in  general  reading  will  no  doubt  involve  decimals 
if.  the  work  is  not  given  too  early.  The  sixth  grade 
seems  the  most  suitable  place  in  the  curriculum  for  the 
subject.  Then  such  relations  as  the  circumference  of  a 
circle  to  its  diameter,  the  amount  of  rainfall,  the  crop's 
yield  per  acre,  the  digestive  nutriments  of  stock  feed  and 
of  human  food,  problems  involving  time  and  speed,  the 
various  constituents  of  fertilizers,  all  these  furnish  a 
motive  for  the  work  in  decimals.  The  use  in  the  subject 
of  percentage,  however,  is  the  most  common  use  that 
the  pupil  will  find  for  the  subject;  and,  hence,  a  study 
of  decimals  should  be  followed  up  by  the  general  dis- 
cussion of  percentage. 


CHAPTER  XI 
PERCENTAGE  AND  ITS  APPLICATIONS 

IN  the  past,  both  textbooks  and  teachers  have  pre- 
sented the  subject  of  percentage  as  if  it  were  some  new 
topic  with  its  own  rules  and  processes,  involving  new 
mathematical  principles.  The  subject  was  not  only 
presented  as  a  new  phase  in  the  development  of  arith- 
metic, but  it  was  subdivided  into  "cases"  each  with  its 
own  rule  and  formula. 

This  method  of  treatment  has  not  entirely  passed.  We 
still  find  textbooks  in  use  that  treat  the  subject  in  this 
way ;  and  many  of  the  older  teachers,  brought  up  by  such 
a  method,  still  persist  in  giving  it  to  their  pupils  as  it  was 
given  them. 

But  such  teaching-  is  rapidly  passing.  The  pupil  is 
led  to  see  that  the  subject  is  merely  a  continuation  of 
the  study  of  fractions,  a  new  language  for  an  old  and  well- 
known  idea,  and  that  this  change  to  a  new  language  does 
not  lead  to  new  processes  or  to  new  principles. 

THE  NOTION  AND  NOTATION  OF  PER  CENT 

The  term  per  cent  must  be  seen  to  be  a  special  name  for 
a  special  fraction,  one  whose  unit  is  hundredths. 

To  introduce  the  need  of  the  new  term,  pupils  may  be 
asked  such  questions  as,  "  If  you  solve  3  out  of  4  of  your 
problems,  what  part  of  them  do  you  solve?"  "If  in  an 

123 


124  THE   TEACHING  OF   ARITHMETIC 

orchard  7  out  of  8  of  the  trees  are  apple  trees,  what  part 
are  apple  trees?"  "If  a  man  in  business  makes  $15  out 
of  every  $100  he  receives  for  goods,  what  part  of  his 
receipts  is  profit?"  etc. 

Lead  the  pupil  to  see  that  it  is  much  easier  to  say  that 
a  merchant  makes  "15  per  cent"  than  to  say  that  he 
makes  "$15  out  of  $100,"  or  to  say  that  he  makes  "15 
hundredths,"  and  that  the  term  "hundredths"  is  rarely 
ever  used  in  business,  but  that  the  term  per  cent,  which 
means  the  same,  is  used  instead.  The  term  is  from  the 
Latin  per  (out  of)  and  centum  (a  hundred).  So  7  per 
cent,  written  7%,  means  7  out  of  100,  or  7  hundredths. 

THE  THREE  PROBLEMS  OF  PERCENTAGE 

All  problems  arising  under  percentage  and  its  applica- 
tions fall  under  three  general  types:  (1)  to  find  a  per 
cent  of  a  number;  (2)  to  find  what  per  cent  one  number 
is  of  another;  and  (3)  to  find  a  number  when  some  per 
cent  of  it  is  known.  Examples  of  these  are:  (1)  Find 
25%  of  300 ;  (2)  75  is  what  per  cent  of  300  ?  and  (3)  75  is 
25%  of  what  number  ? 

The  first  two  of  these  problems  are  by  far  the  most 
common  in  business  practice,  but  the  third  has  a  place, 
as  will  be  shown  later. 

To  FIND  A  PER  CENT  OF  A  NUMBER 

When  the  pupil  understands  the  meaning  of  the  nota- 
tion for  per  cent,  this  class  of  problems  gives  no  difficulty, 
for  he  is  familiar  with  the  process  involved.  He  knows 
that  .06X350  means  .06  of  350,  or  that  .17  of  285  means 


PERCENTAGE  AND  ITS  APPLICATIONS      125 

.17X285;  etc.  So  to  find  18%  of  365  means  .18X365. 
The  only  new  thing,  then,  is  the  notation  and  the  ability 
to  express  any  per  cent  as  a  decimal.  When  the  number 
of  per  cent  is  an  integer  of  but  one  or  two  figures,  as  6% 
or  35%,  the  pupil  has  no  trouble  in  the  translation  to 
hundredths  ;  but  when  the  per  cent  is  a  fraction,  common 
or  decimal,  or  when  it  is  a  whole  number  larger  than  100, 
the  pupil  often  has  difficulty  in  expressing  it  as  hun- 
dredths. Examples  of  such  are:  f%,  .4%,  250%,  etc. 
Errors  in  changing  such  per  cents  to  decimals  are  often 
made  by  normal  school  students  and  by  teachers. 

These  need  not  give  much  trouble  if  properly  pre- 
sented. .4%  means  4  tenths  of  one  hundredth,  hence 
4  thousandths  or  .004.  The  pupil  need  not  go  through 
this  rationalizing  process  for  each,  but  should  visualize  the 
expression  .4%  as  "no  tenths  or  hundredths,  but  4  in  the 
next  lower  order  "  ;  hence,  .4%  =  .004,  .38%  =  .0038,  1  .2%  = 
.012,  etc.,  observing  that,  since  %  is  a  symbol  for  hundredths, 
when  it  is  removed  two  more  decimal  places  must  be  used. 

When  this  type  is  fixed,  it  is  easy  to  present  those 
per  cents  expressed  as  common  fractions.  Thus,  ^%  = 
.5%  =  .005;  f%  =  .75%  =  .0075;  f%  =  .375%  =  .00375. 

The  pupil  will  easily  understand  the  meaning  of  those 
per  cents  larger  than  100  if  they  are  first  expressed  in 
the  common  fraction  form.  Thus, 


And  thus  he  sees  that,  when  removing  the  per  cent  sign, 
two  more  decimal  places  must  be  pointed  off.    So  375%  = 
3.75;  560%  =  5.60  =  5.6. 
It  is  imperative  that  the  pupil  can  express  any  per  cent 


126  THE   TEACHING  OF  ARITHMETIC 

as  a  decimal  before  he  can  find  a  given  per  cent  of  a  num- 
ber; for  to  find  1|%  of  $3800  is  to  find  .0175X$3800; 
to  find  175%  of  396  is  to  find  1.75X396. 

To  find  a  per  cent  of  a  number,  then,  no  new  rule  or 
formula  should  be  used;  but  the  per  cent  should  be 
changed  to  a  decimal  form  and  used  as  a  multiplier. 

To  FIND  WHAT  PER  CENT  ONE  NUMBER  Is  OF  ANOTHER 

In  this  problem  we  are  simply  asking  for  the  relation 
of  two  numbers  when  expressed  as  hundredths  or  per 
cent.  This  involves  the  ratio  idea  of  a  fraction ;  that  is, 
the  use  of  a  fraction  to  express  a  relation,  and  the  ability 
to  change  a  fraction  to  a  decimal.  A  review,  then,  of 
these  two  problems  should  be  given  to"  see  that  the  pupil 
has  the  necessary  ability  to  solve  this  problem  of  per- 
centage. Thus,  if  a  merchant  made  $24  from  a  sale  of 
$96,  he  made  $24  out  of  $96  or  ff  of  the  receipts.  f£  = 
.25  =  25%.  And,  in  general,  to  find  what  per  cent  one' 
number,  as  325,  is  of  another,  as  468,  proceed  as  follows : 

The  steps  are : 

1.  Express  the  relation  as  a  .6944  =  69.44% 
fraction.                                            468)325.00 

2.  Change  the  fraction  to  a  2808 
decimal.  44  20 

3.  Express    the    decimal    in  42 12 
terms  of  per  cent.  2  080 

This  second  problem  is  often  1  872 

treated  as  an  indirect  problem,  208 

the  inverse  of  the  first  problem.  Thus,  to  find  the  result 
in  the  problem  given  above,  the  reasoning  is  that  468 


PERCENTAGE   AND   ITS   APPLICATIONS      127 

multiplied  by  the  answer,  if  known,  would  give  325. 
So  the  statement  is  made  : 

3X468  =  325, 
z  =  325 -*- 468  =  .6944 = 69.44%. 

But  this  method  takes  away  the  relation  idea  expressed 
by  per  cent,  is  more  difficult  to  understand,  and  is  not 
related  to  the  processes  and  principles  already  known, 
and  is  not  to  be  recommended. 

It  will  be  observed  that  this  problem  requires  the 
ability  to  express  any  decimal  as  a  per  cent,  the  inverse 
of  the  ability  needed  in  the  first  problem.  So  some 
practice  in  expressing  such  relations  as  the  following  is 
needed : 

.26  =26%  1.25  =125% 

.07  =7%  .004 =.4% 

.065  =  6.5%  1 .645  =  164.5% 

To  FIND  A  NUMBER  WHEN  A  PER  CENT  OF  IT  Is  KNOWN 

This  type,  usually  called  the  indirect  problem  of  per- 
centage, occurs  less  often  than  the  other  two.  It  is  quite 
commonly  urged  in  current  articles  on  the  teaching  of 
arithmetic  that  this  type  of  problem  be  omitted  alto- 
gether, and  the  recommendation  would  be  well  founded 
if  there  were  no  other  applications  of  it  than  those  given 
in  most  textbooks.  The  usual  type  given  is,  "If  a  mer- 
chant sold  a  suit  for  $24,  thereby  making  20%  of  the 
cost,  find  the  cost."  Now,  this  is  not  a  problem  that 
meets  the  needs  of  any  one.  The  answer,  if  the  problem 
is  real,  must  have  been  known  before  the  problem  could 


128  THE   TEACHING  OF   ARITHMETIC 

have  been  made.  This  is  a  sort  of  "hide-and-go-seek" 
problem  given  for  "analysis"  or  "mental  discipline"  — 
reasons  no  longer  sufficient  when  so  many  real  problems 
are  to  be  found  on  every  hand  that  meet  the  needs  of 
man  in  doing  the  world's  work. 

The  indirect  problem,  however,  frequently  arises  hi 
actual  business  practice.  Thus,  a  manufacturer,  knowing 
the  cost  of  an  article  and  the  average  overhead  charges, 
etc.,  based  upon  the  sales,  knows  that  the  very  minimum 
price  at  which  the  article  must  be  listed  must  give  a 
certain  per  cent  of  profit  reckoned  on  the  sales. 

Thus,  if  it  costs  $10.26  to  make  an  article,  at  what 
must  it  be  sold  to  give  a  profit  of  24%  of  the  selling  price  ? 

Since  the  cost  and  profit  make  up  the  selling  price,  if 
24%  of  it  is  profit,  the  remaining  76%  of  it  must  be  the 
cost.  So,  if  the  selling  price  of  it  were  known  and  mul- 
tiplied by  .76,  the  result  would  be  the  cost,  or  $10.26. 
Hence,  the  relation : 

.76  X  selling  price  =  $10.26. 

Here  we  have  the  product  of  two  factors  given  ($10.26) 
and  one  of  the  factors  (.76)  to  find  the  other  (selling  price). 
Hence,  it  is  a  case  of  division  from  the  definition  of  divi- 
sion. The  solution  then  is : 

$13.50 
.76)$10.26A 
76 
266 
228 
380 


PERCENTAGE   AND  ITS  APPLICATIONS      129 

Another  real  application  of  the  same  problem  would  be 
to  find  a  catalogue  price  that  must  be  put  upon  an  article 
in  order  to  discount  it  at  a  per  cent  and  yet  give  a  fixed 
net  price.  Thus,  a  merchant  may  know  that  $4.80  is 
the  minimum  price  at  which  he  can  sell  an  article,  but  he 
must  put  a  price  upon  it  from  which  he  can  allow  his 
customer  a  discount  of  20%  and  then  get  $4.80. 

Since  he  gets  but  80%  of  the  list  price,  the  known 
relation  is  .8  X  list  price  =  $4.80;  hence,  the  list  price  is 
$4.80  -5- .8  or  $6. 

In  treating  this  type,  then,  the  problems  should  be 
those  of  the  business  world  and  not  those  now  found  in 
most  textbooks,  whose  only  excuse  for  being  there  is  that 
they  serve  as  a  sort  of  mental  gymnastics. 

THE  APPLICATIONS  OF  PERCENTAGE 

The  problems  given  under  the  topic  of  percentage  are 
for  two  general  purposes:  (1)  to  develop  the  ability  to 
see,  express,  and  interpret  the  relationships  between  any 
magnitudes  in  terms  of  per  cent  as  encountered  in  general 
reading  about  increases,  decreases,  discounts,  etc. ;  and 
(2)  to  give  a  social  insight  into  certain  phases  of  com- 
mercial life  needed  for  a  proper  understanding  of  ref- 
erences met  in  general  reading  and  in  conversation. 

ABILITY  TO  EXPRESS  AND  INTERPRET  RELATIONSHIPS 

IN  PER  CENT 

This  ability  is  developed  through  problems  of  a  general 
nature  relating  to  magnitudes  of  all  kinds  familiar  to  the 
pupil  and  not  confined  to  money  transactions  as  in  buy- 


130  THE    TEACHING  OF   ARITHMETIC 

ing,  selling,  loaning,  etc.  So  often  a  pupil  gets  a  notion 
that  per  cent  refers  to  money  as  "cents  on  a  dollar,"  or 
some  such  false  notion. 

Hypothetical  problems,  both  direct  and  indirect,  may 
be  used  for  such  development  as  long  as  they  do  not 
misrepresent  facts  or  give  a  wrong  notion  of  real  practices 
or  conditions. 

All  three  problems  may  be  brought  out  from  the  same 
data ;  as,  "  The  population  of  a  certain  city  was  390,300 
in  1910  and  468,360  in  1915.  (a)  Find  the  rate  of  in- 
crease in  5  years.  (6)  At  the  same  rate  of  increase,  what 
will  the  population  be  in  1920?  (c)  If  the  population 
increased  at  the  same  rate  from  1905  to  1910,  what  must 
it  have  been  in  1905?" 

BUSINESS  APPLICATIONS  OF  PERCENTAGE 

The  commercial  problems  given  should  be  more  for 
the  purpose  of  giving  a  social  insight  into  current  practices 
in  order  that  one  may  understand  references  to  them  met 
in  general  reading  and  in  conversation,  rather  than  to 
fit  for  a  certain  vocation.  And  yet  the  methods  used 
should  be  those  used  in  business  and  not  some  stereotyped 
forms  of  analysis  and  procedure  so  often  found  in  the 
schoolroom.  The  problems,  too,  must  be  those  met  in 
actual  business  and  made  from  data  fairly  true  to  life 
in  order  to  give  true  information  and  not  misinformation. 

COMMISSION 

As  a  fee  for  services,  salesmen  of  almost  all  lines  of 
goods  get  a  per  cent  of  their  sales,  sometimes  in  addition 


PERCENTAGE   AND   ITS  APPLICATIONS      131 

to  a  fixed  salary  and  sometimes  as  their  only  fee  for  their 
work.  This  fee  is  called  their  commission.  Buyers  also 
usually  work  on  the  commission  or  commission  plus  salary 
basis.  This  application,  then,  is  only  the  first  problem 
of  percentage  and  requires  multiplication  only.  Ex- 
amples of  commission  are  found  on  every  hand.  The 
real  estate  agent  gets  a  per  cent  of  his  sales  as  a  com- 
mission, the  agents  who  canvass  from  house  to  house, 
insurance  agents,  and  salesmen  all  get  a  per  cent  of  their 
sales. 

The  pupil  should  get  clearly  this  meaning  of  commis- 
sion, which  is  the  meaning  he  needs  in  order  to  interpret 
references  met  in  general  reading  or  conversation.  This 
may  be  followed  by  the  work  of  the  commission  merchant 
who  sells  farm  produce,  grain,  cotton,  pork,  etc. 

Indirect  problems  found  in  most  of  the  books  of  a 
decade  or  so  ago  should  be  entirely  omitted,  as  they  give 
a  false  notion  of  business  transactions.  The  following 
is  a  type  of  such  problems :  An  agent  received  $1200 
with  which  to  buy  goods  after  deducting  a  10%  commission 
for  buying.  Find  the  commission  and  the  amount  spent 
for  goods. 

This  gives  a  wrong  notion  of  business  practices,  for  a 
business  firm  would  not  send  its  agent  a  fixed  amount  from 
which  to  buy  goods  after  taking  out  his  fee  for  services. 

DISCOUNT 

In  general  conversation  and  reading,  the  term  discount 
is  used  to  denote  a  deduction  from  a  former  price.  Thus, 
a  boy  may  sell  his  bicycle  at  a  discount  upon  what  he  paid 


132  THE    TEACHING  OF   ARITHMETIC 

for  it  because  he  has  used  it  a  season.  A  store  may  give 
a  discount  on  goods  soiled  or  out  of  season.  A  dealer 
may  allow  a  discount  for  a  cash  payment  from  the  price 
at  which  the  goods  are  sold  on  time.  The  teacher  should 
find  such  familiar  uses  of  the  term  as  the  pupil  meets  in 
his  everyday  life.  The  chief  problem  here,  as  in  commis- 
sion, is  the  first  problem  of  percentage  —  to  find  a  per 
cent  of  a  number.  A  question,  however,  may  arise  as  to 
what  per  cent  a  certain  fixed  discount  is.  For  example, 
a  table  marked  at  $30  may  sell  for  $24  during  a  special 
sale.  What  per  cent  of  discount  was  given  ? 

The  indirect  problem  can  answer  no  real  need,  for  it 
answers  no  question  that  would  arise  either  in  the  mind 
of  the  buyer  or  the  seller,  and  hence  it  should  not  be  given 
under  the  subject  of  discounts.  v 

Commercial  discount  is  a  more  specialized  term  and 
refers  to  transactions  between  the  retailer  and  the  whole- 
saler. But  the  term  has  a  general  interest  to  the  average 
person  and  should  be  considered.  Thus,  it  is  a  custom 
among  certain  classes  of  wholesalers  and  manufacturers 
to  publish  catalogues  with  a  fixed  list  price  of  their  goods 
and  then  allow  a  deduction  to  the  retailer  handling  their 
kinds  of  goods.  This  is  what  is  known  as  commercial 
discount,  in  distinction  from  the  general  use  of  the  term. 

Usually  the  list  price,  or  catalogue  price,  remains  the 
same  for  long  periods;  but,  when  the  market  changes, 
new  discounts  are  made.  If  the  market  price  increases, 
a  smaller  discount  is  allowed;  but,  if  the  market  price 
decreases,  the  discount  is  increased.  The  increase  in 
discount  is  usually  made  by  stating  a  new  discount  to  be 


PERCENTAGE   AND  ITS  APPLICATIONS      133 

applied  to  the  previous  net  price.  When  two  or  more 
discounts  to  be  deducted  in  this  way  are  allowed,  they 
are  called  successive  discounts. 

PROFIT  AND  Loss 

In  the  commercial  \forld  the  subject  of  profit  and  loss 
is  one  of  the  important  applications  of  percentage.  But 
for  the  average  person,  whether  engaged  in  business  or 
not,  the  terms  are  so  much  used  that  the  subject  becomes 
one  of  general  importance. 

This  application  of  percentage  may  properly  bring  in 
all  three  of  the  problems  of  percentage.  Thus,  a  busi- 
ness man  may  be  confronted  with  any  one  of  the  following 
questions : 

(a)  An  article  cost  me  $48.  At  what  price  must  I  sell 
it  to  make  a  gross  profit  of  20%  of  the  cost  ? 

Solution.    $48+20%  of  $48  =  $57.60. 

(6)  An  article  cost  me  $48  and  I  sold  it  for  $57.60. 
What  per  cent  of  the  cost  was  my  gross  profit?  What 
per  cent  of  the  selling  price  ? 

Solution.  $57.60 -$48  =  $9.60,  gross  profit ; 
9.60  -r-  $48  =  20%,  rate  on  cost ; 
9.60  -T- $57.60  =  16f%,  rate  on  sales. 

(c)  An  article  cost  me  $48.  At  what  price  must  I  sell 
it  to  make  16f%  of  the  selling  price? 

Solution.  Since  the  gain  is  £  of  the  selling  price,  the 
cost  is  £  of  it.  Hence,  £  of  the  selling  price  =  $48.  The 
whole  of  it=£X$48  =  $57.60. 


134  THE   TEACHING  OF  ARITHMETIC 

It  must  be  remembered  that,  in  teaching  this  or  any 
other  practical  application  of  arithmetic,  the  purpose  is 
to  give  accurate  information  so  that  one  may  properly 
interpret  business  methods  and  practices.  Thus,  in 
teaching  profit  and  loss  one  should  not  teach  that  loss  or 
gain  is  always  reckoned  on  the  cost  —  a  thing  taught 
in  nearly  all  textbooks  —  for  it  is  not  true.  There  is  no 
uniform  agreement  among  business  men  as  to  what  should 
be  used  as  the  basis  in  reckoning  the  rate  of  profit.  Some 
reckon  the  profit  on  the  net  cost,  some  on  the  delivered 
cost,  and  others  upon  the  selling  price.  No  confusion 
arises,  however,  if  the  basis  is  stated.  But  the  pupil 
must  be  led  to  see  that  the  mere  statement  that  a  mer- 
chant made  25%  has  no  meaning  unless  the  basis  upon 
which  it  was  reckoned  is  stated.  Thus,  one  should  say, 
"he  made  25%  of  the  prime  or  net  cost,"  "25%  of  the 
delivered  cost,"  or  "25%  of  the  sales."  Most  textbooks 
make  this  error.  That  is,  they  still  give  such  problems 
as,  "A  merchant  paid  $24  for  an  article.  For  how  much 
must  he  sell  it  to  gain  20%  ?  "  This  problem  is  not  definite. 
If  one  interprets  the  gain  as  reckoned  on  the  cost,  the 
solution  is:  1. 20 X $24  =  $28.80;  if  it  is  reckoned  upon 
the  selling  price,  the  solution  is:  $24-7- .80  =  $30.  So  it 
should  be  clear  that,  if  a  per  cent  is  to  have  a  definite 
meaning,  it  must  be  followed  by  a  phrase  showing  the 
basis  upon  which  it  is  reckoned. 

SIMPLE  INTEREST 

The  topic  of  simple  interest  is  so  common  in  conversa- 
tion and  in  general  reading  that  it  is  one  of  the  earliest 


PERCENTAGE  AND  ITS  APPLICATIONS      135 

applications  of  percentage  to  be  discussed.  It  presents 
no  new  difficulty  and  the  time  element  is  the  only  feature 
not  already  understood.  The  topic  is  taken  for  informa- 
tion rather  than  to  develop  skill  in  calculating  interest 
or  to  develop  mathematical  power.  So  it  is  not  advisable 
to  develop  special  methods  of  computation  but  rather  to 
teach  a  general  method  that  follows  from  the  meaning  of 
interest.  Hence,  the  pupil  should  understand  that  interest 
is  money  paid  for  the  use  of  money  or  for  an  accommoda- 
tion on  an  unpaid  debt ;  and  that  it  is  reckoned  as  a  per 
cent  of  the  debt  for  a  year's  use  of  it,  even  though  the 
interest  is  collected  semiannually,  quarterly,  or  oftener. 
Thus,  the  interest  of  $600  at  6%  is  $36  for  a  whole  year. 
But  for  6  months  it  would  be  but  half  as  much,  or  $18, 
and  for  but  one  month  it  would  be  $3.  In  this  way  it 
is  easily  shown  that  the  interest  for  one  year  multiplied 
by  the  time  expressed  as  a  part  of  a  year  gives  the  interest 
for  the  given  tune. 

When  the  interest  is  for  some  simple  fractional  part 
of  a  year,  as  to  find  the  interest  of  $1200  at  5%  for  4 
months,  the  work  may  be  arranged  as  follows : 

The  interest  for  1  year  being  5%  of  $1200 
is  $60.    But  for  4  months  the  interest  would         $1200 
be  but  i  of  $60.  .05 

Where  the  fractional  part  of  a  year  is  not      — — : — 

$20  00 
so  simple,  all  the  work  to  be  done  should  first 

be  written  down  and  all  common  factors  can- 
celed.    Thus,  to  find  the  interest  on  $1500  at  6%  for  86 
days,  the  work  should  be  : 


136  THE   TEACHING  OF   ARITHMETIC 


The  only  problem  of  arithmetic  that  concerns  the 
person  who  is  borrowing  or  loaning  money  is  how  to  find 
the  interest.  Hence,  the  first  general  problem  of  per- 
centage is  the  only  one  of  the  three  that  is  used  in  interest. 
The  three  possible  indirect  problems  still  given  by  some 
textbooks  "for  analysis"  should  have  no  place  in  the 
course,  as  they  never  occur  in  life  and  detract  from  the 
real  purpose  of  teaching  the  subject,  which  is  to  develop 
a  social  insight  into  methods  of  borrowing  and  loaning 
money. 

More  important  than  the  mere  ability  to  find  interest 
is  a  knowledge  of  current  rates  of  interest,  the  form  and 
meaning  of  a  promissory  note,  the  methods  of  securing 
payment,  how  often  the  interest  is  usually  collected,  etc. 
While  such  knowledge  is  found  in  most  textbooks,  those 
in  use  may  have  been  written  many  years  ago  and  some 
changes  may  have  taken  place.  So  it  creates  more 
interest  and  insures  accurate  local  information  if  a  com- 
mittee from  the  class  can  go  to  some  bank  and  get  the 
information  and  bring  it  fresh  to  class. 

Those  problems  of  the  textbook  giving  rates  not  in 
use  and  with  the  time  element  more  than  one  year  should 
not  be  given,  for  they  give  wrong  notions  as  to  the  price 
paid  for  money  and  also  as  to  the  time  when  interest  must 
be  paid.  Have  pupils  see  that  a  man  may  give  a  note  for 


PERCENTAGE  AND  ITS  APPLICATIONS      137 

a  loan  and  secure  it  properly,  say  by  a  mortgage  on  real 
estate,  and  not  pay  the  debt  for  years ;  but  that  the  in- 
terest must  be  kept  paid  up  as  it  falls  due,  which  is  at 
least  yearly  if  not  more  often,  as  semiannually,  this 
being  specified  in  the  note. 

SHORT  METHODS  OF  COMPUTING  INTEREST 

In  any  business  in  which  much  computation  of  interest 
is  required,  as  in  banks,  trust  companies,  or  life  insurance 
offices,  use  is  made  of  interest  tables  which  greatly  lessen 
the  labor  of  computation.  A  person  outside  of  such  a 
vocation  has  so  little  need  of  computing  interest  that  it 
seems  unwise  to  teach  more  than  the  general  method  to 
pupils  in  the  grammar  school. 

By  the  general  method  is  meant:  time X rate X 
principal  =  interest,  whether  a  year's  interest  is  first  found 
and  that  result  multiplied  by  the  number  of  years,  or 
when  all  the  work  to  be  done  is  written  and  common 
factors  canceled. 

Since  most  textbooks  give  several  short  methods,  two 
are  given  here.  They  follow  from  first  solving  the  prob- 
lem by  the  general  method,  and  thus  they  help  the  pupil 
to  see  how  special  cases  may  lead  to  special  methods. 

Thus,  to  find  the  interest  of  $1750  at  6%  for  119  days : 

By  the  general  method,  ^X-^-X$1750=  119x$175° 


100  6000 

60 

Interpreting  the  result,  we  see  the  work  to  be  done  may 
be  expressed  as  follows :    To  find  the  interest  of  any 


138  THE   TEACHING  OF  ARITHMETIC 

principal  at  6%  for  any  number  of  days,  multiply  the 
principal  by  the  number  of  days,  point  off  three  more 
decimal  places,  and  divide  by  six. 

Illustration.    Find    the    interest    of    $850         $850 
at  6%  for  70  days.    The  work  is  given  in  70 


the  margin.  6)$59.5QO 

Had  the  number  of  days  been  60,  then  all          $9.92 
factors  would  have  canceled  except  the  prin- 
cipal divided  by  100.    Thus,  to  find  the  interest  of  $1350 
at  6%  for  60  days,  the  general  method  is  : 


100 


From  this  the  pupil  may  observe  that  to  find  the  interest 
of  any  principal  at  6%  for  60  days,  point  off  two  more 
decimal  places  in  the  principal. 

COMPOUND  INTEREST 

When  the  interest  due  at  the  end  of  any  interval  is 
added  to  the  debt  and  thus  draws  interest  for  the  next 
succeeding  interval,  as  in  a  savings  bank,  the  interest 
that  accrues  is  called  compound  interest. 

The  subject  is  of  chief  importance  to  large  investors, 
as  building  and  loan  associations,  life  insurance  companies, 
banking  corporations,  etc.,  who  wish  to  compute  the  final 
incomes  from  reinvesting  all  interest  as  it  falls  due. 
Such  computations  are  made  by  the  use  of  compound 
interest  tables. 


PERCENTAGE   AND   ITS  APPLICATIONS      139 

In  teaching  the  subject  the  teacher  may  use  it  as  the 
basis  of  a  discussion  of  thrift,  the  nature  and  importance 
of  the  savings  bank,  etc.  It  is  always  interesting  to  see 
how  rapidly  small  regular  deposits  increase  at  even  a 
small  rate  of  interest.  Thus,  $1  per  week  for  40  years 
at  5%,  compounded  annually,  will  amount  to  about 
$6600.  And  $1  per  week  for  only  10  years  at  5%,  com- 
pounded annually,  will  amount  to  nearly  $700. 

INSURANCE 

The  subject  of  insurance,  both  property  insurance  and 
health  and  life  insurance,  is  such  an  important  factor  in 
modern  life  that  it  should  be  one  of  the  applications  in- 
cluded under  percentage,  although  the  term  per  cent  is 
not  very  largely  used  in  discussing  insurance.  The  subject 
is  studied,  however,  for  its  informational  value  rather 
than  for  the  arithmetical  problem  involved.  Canceled 
or  specimen  policies  should  be  brought  before  the  class 
and  the  subject  thus  made  as  real  and  vital  as  possible. 

FIRE  INSURANCE 

While  there  are  various  kinds  of  property  insurance, 
as  tornado,  lightning,  burglary,  live  stock,  marine,  plate 
glass,  transit,  etc.,  the  subject  of  fire  insurance  is  the 
most  common  kind  in  most  communities  and  may  be 
taken  up  as  a  type  of  all. 

Fire  insurance  is  an  agreement  by  a  fire  insurance 
company  to  indemnify  the  insured  against  actual  losses 
from  accidental  fires.  The  "loss  by  fire"  includes  any 


140  THE   TEACHING  OF  ARITHMETIC 

damage  resulting  from  chemicals  or  water  used  in  ex- 
tinguishing the  fire. 

The  form  of  policy  is  regulated  by  the  laws  of  the  various 
states.  It  is  a  fixed  form  of  agreement  to  do  or  not  to  do 
certain  things  for  a  fixed  consideration  called  the  premium. 

To  some  policies  are  attached  riders  granting  certain 
privileges  or  making  certain  restrictions.  Among  the 
standard  riders  the  most  important  is  the  average  or  co- 
insurance clause.  This  is  an  agreement  on  the  part  of 
the  insured  to  carry  a  certain  amount  of  insurance  upon 
his  property  or,  failing  to  do  this,  to  become  a  coinsurer 
with  the  company  for  whatever  amount  his  insurance 
lacks  of  the  amount  agreed  upon.  Under  a  coinsurance 
clause,  then,  the  insurance  company  pays  only  such  a 
part  of  any  loss  as  the  amount  of  their  policy  bears  to  the 
amount  of  insurance  agreed  upon.  Thus,  if  a  man  ac- 
cepts an  80%  coinsurance  clause  as  a  part  of  his  contract 
upon  property  valued  at  $10,000,  he  agrees  to  carry 
$8000  of  insurance.  If  he  carries  but  $6000  and  a  loss 
occurs,  he  can  collect  but  three  fourths  of  it  from  the 
insurance  company. 

The  rates  of  insurance  depend  upon  the  nature  of  the 
risk.  Upon  a  building  they  depend  very  largely  upon : 
(1)  the  location ;  (2)  the  construction ;  (3)  the  occupancy ; 
(4)  the  exposure;  and  (5)  whether  or  not  there  is  co- 
insurance. The  pupils  should  find  the  rates  upon  build- 
ings in  various  parts  of  the  community  and  attempt  to 
justify  the  different  rates.  Details  of  local  conditions 
should  be  learned  from  some  agent  in  the  neighborhood 
and  the  problems  made  to  meet  these  conditions. 


PERCENTAGE  AND  ITS  APPLICATIONS      141 

LIFE  INSURANCE 

About  the  only  arithmetical  problem  of  life  insurance 
that  a  pupil  in  the  grades  is  able  to  handle  is  to  find  the 
difference  in  the  amounts  left  the  beneficiary  by  life 
insurance  and  the  amounts  of  the  premiums  at  simple 
or  compound  interest,  knowing  the  number  of  premiums 
that  have  been  paid.  But  he  can  understand  the  nature 
of  the  three  general  forms  of  policies  and  .also  the  ele- 
ments that  make  up  the  premium. 

The  three  general  forms  of  policies  are:  (1)  ordinary 
life ;  (2)  limited  life ;  and  (3)  endowment. 

In  the  ordinary  life  policy,  the  premiums  are  paid 
during  the  life  of  the  insured,  and  the  insurance  company 
agrees  to  pay  a  fixed  sum  to  the  beneficiary  at  the  death 
of  the  insured. 

In  the  limited  life  policy,  the  premiums  are  paid  for 
a  fixed  number  of  years,  but  the  guarantee  in  the  policy 
is  not  paid  until  the  death  of  the  insured. 

In  an  endowment  policy,  the  insurance  company  agrees 
to  pay  the  insured  the  full  amount  of  the  policy  if  he  sur- 
vives beyond  a  specified  date,  or  to  pay  the  amount  as 
above  if  he  dies  before  that  date. 

The  premium  that  the  insured  pays  is  composed  of 
three  items:  (1)  mortality  cost;  (2)  reserve;  and  (3)  ex- 
pense loading.  The  first  two  of  these  items  form  the  net 
premium  and  are  determined  by  a  given  mortality  table. 
The  American  Mortality  Tables  are  compiled  from  the 
experience  of  the  New  York  Mutual  Life  Insurance 
Company  and  are  the  tables  prescribed  by  statute  in 


142  THE   TEACHING  OF  ARITHMETIC 

most  of  the  states  as  the  basis  upon  which  the  reserve  of 
life  insurance  companies  shall  be  computed. 

The  mortality  cost  is  the  amount  necessary  to  collect 
at  the  beginning  of  each  year  to  pay  the  death  claims  of 
the  current  year  as  determined  by  the  mortality  tables. 

The  reserve  element  of  a  premium  is  the  amount  set 
aside  from  each  premium  at  a  given  rate  of  compound 
interest,  usually  3%  or  3^%,  which  will  amount  to  the 
face  of  the  policy  in  a  given  time.  So  the  company's 
risk  at  any  time  is  only  the  difference  between  the  face 
of  the  policy  and  the  amount  of  the  reserve. 

The  expense  loading,  which  is  from  15%  to  25%  of  the 
net  premium,  is  added  to  meet  the  expenses  of  manage- 
ment. 

While  there  are  other  important  things  to  consider  in 
life  insurance,  the  things  discussed  here  are  sufficient  to 
show  a  pupil  the  types  of  policies  and  the  things  for  which 
the  premiums  must  be  collected.  The  problems  of  the 
subject  are  too  difficult  for  a  pupil  in  the  grammar  school. 
The  subject,  then,  is  for  information  and  not  for  the 
mathematics  involved. 

TAXES 

The  subject  of  taxes,  again,  is  taken  for  its  informa- 
tional value,  not  its  mathematical  problems.  The  pupil 
should  understand  for  what  purposes  taxes  are  levied, 
the  way  in  which  the  expenses  to  run  a  town,  county,  or 
state  are  equitably  distributed,  and  the  current  rates  of 
taxation.  Since  the  method  of  levying  and  collecting 
taxes  varies  in  different  states,  the  local  problems  and 


PERCENTAGE   AND  ITS  APPLICATIONS      143 

rates  are  to  receive  the  chief  emphasis.  In  such  a  study 
of  the  local  methods,  the  pupil  should  become  acquainted 
with  such  terms  as  assessment,  valuation,  assessor,  col- 
lector, board  of  review,  board  of  equalization,  and  delin- 
quent taxes. 

The  tax  rates  are  usually  expressed  as  mills  on  a  dollar, 
dollars  on  a  hundred  dollars,  or  dollars  on  a  thousand 
dollars.  The  general  method  of  finding  the  rate  is  to 
divide  the  amount  to  be  raised  by  the  assessed  valuation 
of  the  property. 

To  facilitate  computation,  a  tax  table  is  usually  pre- 
pared giving  the  taxes  on  sums  from  $10  to  $99,  and  then 
the  tax  on  other  sums  is  found  by  properly  pointing  off 
and  by  addition.  The  pupils  may  no  doubt  be  able  to 
get  a  copy  of  such  a  table  at  the  office  of  the  assessor  of 
taxes. 

The  pupil  should  see  the  difference  between  collecting 
taxes  to  meet  the  expenses  of  the  city,  county,  and  state, 
and  the  method  of  collecting  those  for  the  expenses  of 
the  National  Government.  This  will  lead  to  a  study  of 
duties  and  customs  and  the  income  tax.  Here  again  the 
mathematics  involved  is  of  less  importance  than  the  in- 
formation about  the  general  expenses  of  the  government, 
where  they  go,  and  how  they  are  met. 

BANKING 

Pupils  should  know  the  function  of  a  modern  bank  — 
that  it  is  an  institution  where  money  is  deposited  for 
safe-keeping  and  from  which  it  may  be  withdrawn  when 
wanted ;  and  also  that  it  is  a  place  where  money  is  loaned 


144  THE   TEACHING  OF  ARITHMETIC 

on  personal  or  other  security.  They  should  know,  too, 
that  most  of  the  money  that  a  bank  handles  is  that  of  its 
depositors,  and  that  it  is  from  loaning  these  deposits  that 
a  bank  earns  money. 

In  a  commercial  sense,  banks  are  classified  as  banks 
of  deposit,  banks  of  discount,  and  banks  of  circulation.  All 
commercial  banks  exercise  the  first  two  of  these  functions, 
and  national  banks  the  third.  The  function  of  a  bank 
of  deposit  is  to  receive  money  on  deposit  and  pay  the 
checks  drawn  upon  these  deposits.  The  function  of  a 
bank  of  discount  is  to  loan  money  on  promissory  notes. 
The  function  of  a  bank  of  circulation  is  to  issue  its  own 
promissory  notes,  which  are  used  as  currency.  To  make 
this  last  clear,  take  before  the  class  some  bank  note 
issued  by  some  national  bank  or  federal  reserve  bank, 
the  only  institutions  allowed  to  issue  them. 

The  pupils  should  also  understand  the  method  of 
opening  an  account  with  a  bank,  the  making  out  of  a 
ticket  of  deposit,  making  out  and  indorsing  a  check,  and 
the  purpose  of  having  a  check,  that  is  to  be  sent  out  of 
town,  certified  by  the  bank  upon  which  it  is  drawn,  and 
the  transmitting  of  money  by  means  of  a  bank  draft. 
All  this  is  vitalized  by  having  pupils  gather  the  informa- 
tion from  a  local  bank. 

BANK  DISCOUNT 

The  subject  of  bank  discount  need  not  present  any  new 
difficulties.  It  is  only  a  method  of  collecting  interest. 
When  interest  is  collected  in  advance,  it  is  called  bank 
discount.  Thus,  if  one  borrows  $300  at  a  bank  for  90 


PERCENTAGE  AND  ITS  APPLICATIONS      145 

days,  the  cashier  computes  the  Interest  at  the  bank's  rate, 
which  is  quite  generally  6%,  and  deducts  it  from  the  §300 
or  face  of  the  note.  At  6%  the  interest  is  $4.50,  leaving 
$295.50,  called  the  proceeds.  When  the  interest  is  col- 
lected in  advance,  a  non-interest-bearing  note  is  given 
for  the  loan,  for  only  the  $300  is  paid  when  the  note  is  due, 
not  $300  "with  interest."  Since  one  paying  interest  in, 
advance  is  paying  upon  the  maturity  value  of  the  note, 
it  follows  that,  in  discounting  an  interest-bearing  note  at 
a  bank,  the  bank  would  reckon  the  interest  on  the  amount 
of  the  note  when  due  and  not  upon  the  face  of  the  note. 
Thus,  if  a  note  of  $1000  bearing  5%  interest  is  to  run  one 
year,  it  is  worth  $1050  when  due.  So,  if  it  is  sold  at  a 
bank  60  days  before  it  is  due,  the  bank  will  compute  the 
interest  on  $1050  for  60  days,  and  not  upon  $1000.  The 
interest  (bank  discount),  then,  is  $10.50,  leaving  the 
proceeds  of  $1039.50. 

STOCK  INVESTMENTS 

In  the  past,  the  applications  of  percentage  have  been 
hard  for  the  pupil  in  the  grammar  school  for  two  reasons : 
(1)  more  time  has  been  spent  upon  the  mathematical 
side  than  upon  the  social  side,  hence  the  pupil  has  not  had 
a  concrete  background  for  the  arithmetical  side  of  the 
work;  and  (2)  the  indirect  problem,  never  arising  in 
business,  has  been  featured  for  "analysis."  If  teachers 
and  textbooks  would  give  merely  the  problems  needed  to 
interpret  the  references  met  in  general  reading  and  con- 
versation, the  work  would  not  offer  great  difficulties. 

Interest  in  the  stock  market  is  so  general  that  the 


146  THE   TEACHING  OF  ARITHMETIC 

subject  becomes  one  of  the  important  applications, 
but  for  its  information  and  not  for  the  mathematics 
involved. 

There  was  a  time  when  the  owner  of  a  business  took  hi 
one  or  more  partners  when  he  wished  to  increase  the  busi- 
ness beyond  his  own  power  to  supply  the  necessary  capital. 
But  such  partnerships  often  proved  unsatisfactory.  Each 
partner  was  usually  held  responsible  for  the  acts  of  any 
one  of  the  partners.  Each  change  of  partnership  required 
new  adjustments.  As  business  enlarges,  these  and  many 
other  difficulties  arise. 

Organizing  business  under  a  corporation  or  company 
does  away  with  many  of  the  objections  of  the  partner- 
ship arrangement.  Some  of  the  advantages  of  a  corpora- 
tion are:  (1)  one  may  become  but  a  small  investor  in 
several  corporations,  giving  no  personal  attention  to  the 
business,  instead  of  having  to  make  a  large  investment 
and  giving  the  business  more  or  less  attention  as  in  the 
case  of  partnership;  (2)  one  may  transfer  his  ownership 
in  a  corporation  without  affecting  the  business;  (3)  the 
business  can  more  easily  expand  its  capital ;  and  (4)  the 
plan  enables  the  hiring  of  experts  to  run  the  business 
rather  than  having  it  run  directly  by  the  partners. 

In  organizing  a  corporation  a  special  number  of  persons 
interested  in  starting  a  business  of  some  sort  make  ap- 
plication to  the  Secretary  of  State  of  their  own  or  some 
other  state,  giving  the  proposed  name  of  the  company, 
the  place  and  nature  of  the  business  they  propose  to  carry 
on,  the  amount  of  capital  and  the  number  of  shares  into 
which  they  propose  to  divide  it,  and  any  other  informa- 


PERCENTAGE   AND  ITS  APPLICATIONS      147 

tion  required  by  the  law  of  the  state  in  which  they  make 
the  application. 

The  Secretary  of  State  files  their  application  and  grants 
them  a  permit  to  sell  the  capital  stock. 

When  the  legal  amount  has  been  sold,  the  buyers  of 
the  stock  adopt  a  set  of  by-laws  and  elect  a  board  of  di- 
rectors who  in  turn  elect  a  president,  secretary,  and 
treasurer,  and  other  officers  from  their  number.  A 
record  of  this,  showing  that  the  law  has  been  complied 
with,  is  then  filed  with  the  Secretary  of  State,  who  issues 
a  charter,  which  is  an  instrument  defining  the  powers, 
rights,  and  duties  of  the  corporation. 

The  capital  with  which  the  company  organizes  divided 
by  the  number  of  shares  into  which  it  has  seemed  de- 
sirable to  divide  it  gives  the  par  value  of  a  share.  The 
par  value,  then,  is  no  indication  of  the  real  value  of  the 
stock,  but  serves  to  show  what  part  of  the  business  is 
owned  by  the  holder  of  a  certificate.  Thus,  if  the  $100,000 
capital  of  a  corporation  is  divided  into  1000  equal  shares, 
the  size  of  each  is  $100.  So  a  person  owns  one  thousandth 
of  the  business  for  each  share  that  he  owns.  This  simpli- 
fies the  distribution  of  the  earnings. 

The  real  or  market  value  of  stock  is  what  it  can  be 
bought  or  sold  for  in  open  market.  Chief  among  the 
factors  affecting  the  market  price  of  stock  are :  (1)  the 
real  or  prospective  earning  power  of  the  corporation; 
and  (2)  the  confidence  of  the  buying  public,  or  lack  of  it, 
in  the  general  stability  of  the  enterprise.  When  the  real 
or  anticipated  earnings  are  small,  the  price  is  low ;  when 
large,  the  price  is  high. 


148  THE   TEACHING  OF  ARITHMETIC 

The  earnings  of  a  corporation  distributed  among  its 
stockholders  are  called  the  dividends.  They  are  declared  as 
a  per  cent  of  the  capital  of  the  corporation,  and  each  stock- 
holder gets  that  per  cent  of  the  par  value  of  the  stock 
that  he  owns.  Thus,  if  a  corporation,  whose  capital  is 
$200,000,  is  to  distribute  $24,000  in  profits,  the  directors 
declare  a  12%  dividend  and  the  holder  of  each  $100  share 
gets  $12.  Stock  of  this  kind,  whose  dividend  depends 
upon  the  earnings  of  the  corporation,  is  called  common 
stock.  There  is  another  kind  of  stock  sometimes  issued 
called  preferred  stock,  in  which  the  rate  of  dividend 
paid  is  named  in  the  certificate.  These  dividends,  then, 
become  an  obligation  of  the  corporation  and  are  deducted 
from  the  gross  earnings  before  a  dividend  can  be  declared 
upon  the  common  stock. 

There  are  two  types  of  buyers  of  stock:  the  investor 
who  buys  and  holds  for  the  dividends  he  will  receive, 
just  as  he  loans  money  for  the  interest  or  buys  a  house 
for  the  rent;  and  the  speculator  who  buys,  expecting 
the  price  to  advance  so  that  he  can  sell  at  a  profit. 

The  problems  that  naturally  arise  are  very  simple 
when  one  understands  the  terms  involved.  They  are: 
(1)  to  find  how  much  is  lost  or  gained  when  buying  stock 
at  one  price  and  selling  it  at  another;  and  (2)  to  find 
what  per  cent  of  its  market  value  stock  paying  a  certain 
dividend  is  earning.  These  are  the  only  problems  with 
which  the  general  reader  or  the  buyer  is  concerned. 


CHAPTER  XII 
DENOMINATE  NUMBERS  AND  MEASUREMENTS 

I.  NATURE  AND  PRINCIPLES  OF  DENOMINATE 
NUMBERS 

FOEMERLY  the  subject  of  denominate  numbers  was 
taught  as  a  distinct  topic.  Now  the  work  is  distributed 
throughout  the  grades.  The  tables  are  taught  as  need 
for  them  arises  in  the  child's  activities  in  or  out  of  school. 
A  single  unit  from  a  table  may  be  taught  alone  as  need 
for  that  unit  arises  without  teaching  its  relation  to  the 
other  units  of  the  table.  Thus,  the  child  may  recognize 
a  foot  in  length  and  use  the  measure  in  finding  length 
without  knowing  that  12  inches  make  1  foot  or  that  3 
feet  make  1  yard. 

To  get  a  practical  knowledge  of  the  tables  or  the  various 
units,  the  pupil  must  see  and  handle  all  the  common 
measures.  From  these  units  he  should  make  his  own 
table.  Thus  through  measuring  he  discovers  that  2  pt.  = 
1  qt.  and  that  4  qt.  =  1  gal. ;  or  that  12  in.  =  1  ft.  and  that 
3  ft.  =  1  yd.  It  is,  of  course,  unnecessary  and  impractical 
to  have  all  of  a  table  found  in  this  way  where  a  very  large 
number  is  required,  as  finding  how  many  feet  or  yards 
in  a  mile.  After  the  relation  of  the  various  units  of  a 
table  has  been  found,  through  measurement,  the  tables 

149 


150  THE   TEACHING  OF  ARITHMETIC 

should  be  memorized  and  used  in  problems;  but  just  as 
much  of  a  table  as  is  needed  at  the  time  should  be  learned. 
Thus,  inches,  feet,  and  yards  are  needed  long  before  rods 
and  miles  are  needed ;  ounces  and  pounds  before  tons ; 
pints  and  quarts  before  gallons  or  barrels ;  etc. 

All  obsolete  units  of  any  table  and  all  tables  belonging 
exclusively  to  a  technical  education  should  be  entirely 
omitted. 

REDUCTION  OF  DENOMINATE  NUMBERS 

The  reduction  of  denominate  numbers  involves  no  new 
principles.  If  a  pupil  understands  the  meaning  of  multi- 
plication and  division  and  when  to  apply  these  processes 
to  simple  one-step  problems,  he  should  have  no  difficulty 
in  reducing  units  from  one  denomination  to  another  when 
need  for  such  a  reduction  occurs  in  any  practical  applica- 
tion. 

The  old  plan  of  having  each  table  of  the  weights  and 
measures  followed  by  a  lot  of  exercises  for  reductions, 
apart  from  any  industrial  or  commercial  problem,  is  fast 
disappearing,  and  the  reductions  are  now  being  made 
when  need  for  them  arises  in  some  problem  of  everyday 
life. 

Such  reductions,  either  descending  or  ascending,  usually 
involve  one  step,  or  one  step  and  the  addition  of  a  lower 
unit,  or  one  division  and  a  remainder. 

Such  reductions  embody  the  following  operations  : 

(1)  Reduce  5  ft.  8  in.  to  inches. 

5ft.=  5Xl2  in.  =  60  in.;    60  in.+8  in.  =  68in. 


DENOMINATE  NUMBERS  151 

(2)  Reduce  68  in.  to  feet  or  to  feet  and  inches. 
1  in.  =  -&ft;   68  in.  =  68X^r  ft.  =  5|  ft. 

Or,  68  in.  -T-  12  in.  =  5,  plus  remainder  of  8  in. 
not  divided.    Hence,  68  in.  =  5  ft.  8  in. 

That  is,  the  last  solution  is  the  measuring  problem 
of  division.  The  quotient  5  shows  the  number  of  12  in. 
in  68  in. ;  and,  since  each  12  in.  is  a  foot,  5  is  the  number 
of  feet. 

Avoid  such  erroneous  statements  as : 

5  ft.  12)68  in. 
12  5  ft.  8  in. 

60  in.  or 

_8  in.  12  in.)68  in. 
68  in.  5  ft.  8  in. 

ADDITION,  SUBTRACTION,  MULTIPLICATION,  AND  DIVISION 

The  four  fundamental  processes  of  arithmetic  need  not 
be  taught  as  topics  and  processes  of  denominate  numbers 
to  any  great  extent.  But  little,  if  any,  use  of  such  pro- 
cesses occurs  in  the  everyday  problems  of  life.  In  ordinary 
measurement,  results  are  expressed  by  use  of  fractions 
in  terms  of  but  one  unit,  instead  of  two  or  more  before 
adding,  subtracting,  multiplying,  or  dividing  them. 
Thus,  instead  of  adding  or  subtracting  16  ft.  8  in.  and 
12  ft.  6  in.,  one  would  add  or  subtract  16f  ft.  and  12^-  ft. 
Instead  of  multiplying  2  bu.  3  pk.,  one  would  multiply 
2|  bu.  or  2.75  bu.  Instead  of  dividing  12  ft.  10  in.,  one 
would  divide  12f  ft.  or  12.83  ft.,  etc. 


152  THE   TEACHING  OF   ARITHMETIC 

H.   MENSURATION 

If  properly  presented,  the  interpretative  value  of  the 
subject  of  mensuration  is  very  great.  A  large  number 
of  terms  and  concepts  met  in  a  wide  variety  of  practical 
activities  is  acquired.  These  include  the  ideas  of  dis- 
tance, area,  and  volume  as  applied  in  scale  drawing,  in 
computing  costs  in  paving  streets  and  laying  sidewalks, 
in  measuring  farms  and  computing  productions,  in  measur- 
ing bins  for  grains,  coal,  etc.,  in  computing  costs  of  ex- 
cavations; in  fact  on  every  hand  are  found  applications 
of  these  fundamental  facts  of  mensuration.  And  because 
there  is  such  a  rich  field  of  applications,  opening  up  such 
a  possibility  of  problems  of  great  variety,  the  topic,  like 
many  other  topics  of  arithmetic,  is  often  carried  beyond 
its  real  need  as  a  tool  with  which  we  interpret  the  physical 
world  about  us. 

But  to  have  a  broad  interpretative  value  to  the  pupil, 
he  must  be  led  to  formulate  his  own  laws  and  rules  and  to 
find  an  application  for  them  in  the  life  about  him.  It  is 
not  uncommon  to  find  pupils  computing  the  cost  of  side- 
walks of  given  dimensions  as  problems  found  in  their 
textbooks  and  yet  who  have  no  idea  of  the  cost  of  the 
walk  in  front  of  their  own  homes  or  the  school,  or  who  are 
unable  even  to  go  out  and  take  the  measures  and  find  the 
cost  of  such  a  walk.  This  sort  of  teaching  is  valueless 
as  a  means  of  developing  ability  to  interpret  life  about 
them. 

It  is  only  when  the  subject  is  taught  through  having 
the  pupil  make  his  own  measurements,  constructions, 


DENOMINATE  NUMBERS  153 

drawings,  etc.,  and  applying  the  knowledge  gained  to  out- 
of-school  measurements  that  the  subject  becomes  of  real 
value. 

THE  AKEA  OF  A  RECTANGLE 

The  pupil  must  see  that  to  measure  anything  is  simply 
to  apply  some  standard  unit  and  see  how  many  times  it 
is  contained ;  and  thus  to  measure  the  area  of  a  rectangle 
is  to  see  how  many  times  it  contains  a  chosen  square 
whose  side  is  some  linear  unit. 

Through  actual  construction  the  pupil  should  draw  a 
rectangle  whose  sides  are  some  integral  number  of  units 
and  then  lay  it  off  into  square  units  and  count  them  to 
find  the  area  in  order  to  see  clearly  what  is  meant  by  the 
measurement  of  a  rectangle.  Through  drawing  to  a 
scale  and  the  use  of  squared  paper 
upon  which  to  represent  rectangles, 
he  should  become  familiar  with  such 
facts  as :  the  number  of  squares  in  a 
row  along  any  side  is  the  same  as  the 
number  of  linear  units  in  that  side; 
that  there  are  as  many  rows,  all  alike,  as  there  are 
units  in  the  other  dimension;  and  thus  he  should  see 
that  the  total  number  of  square  units  is  the  product  of 
the  number  of  linear  units  in  the  two  dimensions.  Thus, 
the  area  of  a  rectangle  4  inches  by  3  inches  is  4X3  square 
inches. 

Properly  developed,  pupils  will  never  get  the  erroneous 
notion  that  "inches  multiplied  by  inches  give  square 
inches,"  a  thing  still  taught  in  many  schools.  Such  a 


154  THE   TEACHING  OF   ARITHMETIC 

statement  would  seem  as  ridiculous  to  them  as  "  3  horses  X 
4  horses  =  12  square  horses." 

Following  the  development  of  the  rule  or  law,  pupils 
should  apply  their  knowledge  by  measuring  actual  areas 
about  them.  In  finding  costs,  get  local  prices  so  that 
when  the  cost  of  plastering  a  wall,  laying  a  floor,  construct- 
ing a  sidewalk,  paving  a  street,  sodding  a  lawn,  or  roofing 
a  house  is  computed  the  pupil  will  see  about  how  much 
of  the  thing  measured  can  be  bought  for  a  certain  price ; 
that  is,  the  pupil  should  know  what  the  sidewalk  in  front 
of  his  home  or  in  front  of  the  school  would  cost.  He 
should  have  some  idea  of  the  cost  to  plaster  a  wall  of  the 
schoolroom,  or  to  place  the  metal  ceiling  if  there  is  one, 
or  the  cost  of  the  blackboards.  It  is  through  such  work 
as  this  that  the  topic  has  a  real  social  value. 

THE  AREA  OF  A  PARALLELOGRAM 

Through  construction,  the  pupil  must  see  that  a  gen- 
eral parallelogram  whose  angles  are  not  right  angles  can- 
not be  directly  laid  off  into  square  niches  as  in  the  case  of 
the  rectangle.  Then  impress  upon  him  the  fact  that  the 
rectangle,  being  the  only  figure  that  can  be  laid  off  into 
squares,  is  taken  first  and  used  as  a  basis,  and  that  all 
other  figures  are  measured  by  finding  their  relation  to 
some  rectangle. 

The  first  step,  then,  in  measuring  a  parallelogram  is 
to  reduce  it  to  a  rectangle.  The  teacher's  blackboard 
drawings  are  not  conclusive  enough  for  the  child.  The 
part  which  she  removes  by  erasing  and  the  part  which 
she  places  in  another  position  by  drawing  are  not  the 


DENOMINATE   NUMBERS 


155 


same  parts.  From  her  blackboard  presentation  of  the 
method  of  transforming  a  parallelogram  into  an  equivalent 
rectangle,  the  pupil  should  construct  a  parallelogram 
from  cardboard,  actually  cut  off  a  part,  and,  by  rearrang- 
ing this  part,  transform  the  given  parallelogram  into  a 
rectangle.  He  need  not  divide  this  new  rectangle  into 
square  inches,  for  he  can  deduce 
the  rule  from  that  for  finding 
the  area  of  a  rectangle,  for  he 
has  now  discovered  that  the 
area  of  a  parallelogram  is  the 
same  as  that  of  a  rectangle 
having  the  same  dimensions. 
His  attention  should  be  called 
to  the  fact  that  the  perpendicu- 
lar distance  between  the  base  and  opposite  side  becomes 
the  altitude  of  the  rectangle  and  is  thus  the  altitude  of 
the  parallelogram ;  otherwise  he  may  associate  the  two 
adjacent  sides  of  the  parallelogram  with  its  dimensions. 

The  pupils  may  not  be  able  to  find  as  many  examples 
of  this  figure  as  of  the  other  four  plane  areas.  It  is  given 
largely  as  a  basis  for  the  others  in  order  to  save  convert- 
ing them  into  rectangles. 

THE  AEEA  OF  A  TRIANGLE 

As  in  the  parallelogram,  pupils  will  see  that  a  triangle 
cannot  be  divided  directly  into  squares  as  in  the  case  of 
a  rectangle.  After  having  thus  raised  the  question  of 
how  to  measure  a  triangle,  the  teacher  should  have  each 
pupil  cut  two  triangles  of  the  same  form  and  size  from 


156  THE   TEACHING  OF  ARITHMETIC 

cardboard.  But  all  children  should  not  make  triangles  of 
the  same  size.  The  teacher  should  draw  various  forms 
and  sizes  on  the  blackboard  and  let  pupils  use  their  own 
pleasure  in  drawing  any  shape  they  wish.  Then  have 
each  child  place  his  two  triangles  together  so  as  to  form  a 

parallelogram  and  thus  discover 
that  a  triangle  is  just  half  as 
large  as  a  parallelogram  having 
the  same  dimensions.  From  this 
fact,  he  makes  his  rule.  The 
examples  of  this  kind  of  figure  are  more  numerous.  Have 
hun  find  areas  to  be  measured  that  are  triangular  in  form, 
take  the  measurements  himself,  and  find  the  areas. 

THE  AREA  OF  A  TRAPEZOID 

Since  the  pupil  should  be  shown  how  to  find  an  area 
by  dividing  it  up  into  figures  that  he  has  studied,  it  is 
hardly  necessary  to  take  up  the  measurement  of  a  trape- 
zoid for  any  practical  purpose ;  yet  it  is  required  by  most 

courses  of  study  and  

is  so  easy  to  present 
that  the  teaching  of 
it  is  not  open  to  seri- 
ous criticism.  As  in 

the  triangle,  have  the  pupils  each  make  two  trapezoids 
just  alike  and  place  them  so  as  to  form  a  parallelogram. 
They  thus  see  that,  since  it  took  two  trapezoids  to  form 
the  parallelogram,  a  trapezoid  is  half  as  large  as  a  parallelo- 
gram having  a  base  equal  to  the  sum  of  the  bases  of  the 
trapezoid  and  having  the  same  altitude  as  the  trapezoid. 


DENOMINATE  NUMBERS 


157 


As  with  all  other  figures,  have  the  pupils  look  for 
examples  of  such  figures,  measure  them,  and  find  the  areas. 

THE  CIRCUMFERENCE  AND  AREA  OF  A  CIRCLE 
The  pupil  does  not  have  to  accept  the  facts  and  rules 
for  the  measurement  of  a  circle  upon  the  authority  of 
textbook  or  teacher  any  more  than  he  did  the  four  pre- 
ceding areas.  The  rules  and  relations  are  as  readily  dem- 
onstrated objectively,  and  the  pupil  easily  discovers 
them  for  himself. 

In  finding  the  relation  of  the  circumference  to  the 
diameter,  use  as  large  circular  objects  as  possible.  The 
pupils  may  be  directed  to  measure  the  circumference 
and  diameter  of  some  objects  found  at  home,  as  the  din- 
ing room  or  reading  room  tables,  and  bring  the  data  to 
school.  If  the  measurements  have  all  been  carefully 
taken,  the  ratios  of  circumference  to  diameter  in  all  cases 
will  be  nearly  enough  87-  to  satisfy  the  class  that  the 
ratio  is  constant.  The  more  exact  ratio,  7r  =  3.1416, 
should  be  given  and  used  where  great  accuracy  is  required. 


A  B 

To  discover  the  area,  have  pupils  each  cut  from  card- 
board circles  of  various  sizes,  not  too  small,  and  divide 
them  into  at  least  sixteen  equal  parts  as  in  figure  A. 


158  THE   TEACHING  OF  ARITHMETIC 

Have  them  take  half  of  these  parts  and  arrange  them  as 
in  the  lower  half  of  figure  B,  and  then  take  the  other  half 
and  fit  them  into  these,  completing  figure  B,  and  thus 
discover  that  the  area  of  a  circle  is  equal  to  that  of  a 
parallelogram  whose  base  is  half  the  circumference  and 
whose  altitude  is  the  radius.  Then,  A=%CxR.  But 
C=TrD,  hence  %C=irR.  Then,  A=irRxR=7rIP. 
Find  all  the  local  applications  possible. 

THE  VOLUME  OF  RECTANGULAR  SOLIDS 

This  is  the  fundamental  volume  from  which  the  meas- 
urement of  other  volumes  is  deduced.  The  same  plan 
of  comparing  the  volume  with  some  standard  unit,  sug- 
gested hi  the  measurement  of  a  rectangle,  should  be  fol- 
lowed here.  Or,  instead  of  dividing  a  solid  into  cubic 
units,  a  box  whose  inside  measure  is  some  integral  number 
of  units  may  be  filled  with  cubic  units.  The  pupil  should 
see  that  the  number  of  cubic  units  along  any  dimension, 
as  the  length,  is  the  number  of  linear  units  in  the  length ; 
the  number  of  such  rows  in  the  width  is  the  number  of 
linear  units  in  the  width;  and  that  the  number  of  such 
layers  in  the  whole  volume  is  the  number  of  linear  units 
in  the  height.  Thus,  if  there  are  6  inches  in  the  length, 
there  will  be  6  cu.  in.  in  a  row  along  the  length.  If  4 
inches  wide,  then  there  will  be  4X6  cu.  in.  in  a  layer; 
and,  if  5  inches  deep,  there  will  be  in  the  whole  volume 
5X4X6  cu.  in.  or  120  cu.  in.  If  the  rule  is  developed  in 
this  way,  the  pupil  will  not  use  such  erroneous  statements 
as  5  in.X4  in.X6  in.  =  120  cu.  in. 

Follow  up  the  development  by  local  problems,  as  ex- 


DENOMINATE  NUMBERS 


159 


cavations  for  cellars,  capacity  in  tons  of  coal  bins,  in 
bushels  of  grain  bins,  the  air  space  of  the  room,  the  snow 
removed  from  walks,  or  hi  any  grading  or  constructing 
work  that  is  being  done  in  the  neighborhood. 

OTHER  VOLUMES 

The  illustrations  already  given  serve  to  emphasize 
the  method  of  teaching  measurement.  Any  good  text- 
book gives  diagrams  to  illustrate  the  development.  But 
with  children,  diagrams  of  the  textbook  are  not  sufficient. 
The  objects  illustrated  by  the  diagrams  should  be  used. 

GRAPHIC  REPRESENTATION  OF  STATISTICS 

The  relations  between  magnitudes  is  more  easily  grasped 
by  most  people  if  expressed  by  graphs  rather  than  by 
figures  to  represent  the  number  of  units.  For  this  reason 
newspapers,  magazines,  trade  journals,  and  all  kinds  of 
circulars  and  pamphlets  make  very  frequent  use  of  the 


160 


THE   TEACHING  OF  ARITHMETIC 


graph  where  magnitudes  are  shown  in  comparison  with 
other  magnitudes  or  with  their  former  values.  Thus,  the 
graph  on  the  previous  page  is  reproduced  from  an  adver- 
tisement of  the  American  Sugar  Refining  Company. 

There  are  three  general  forms  in  use  depending  upon 
the  nature  of  the  comparisons  to  be  made. 

In  comparing  the  productions,  wealth,  population,  etc., 
of  different  states  or  countries,  straight  lines  are  generally 
used.  Thus,  the  comparative  production  of  corn  in  the 
following  states,  1917,  could  be  represented  graphically 
as  follows : 


Illinois. 
444,843,000  bu. 

Iowa. 
411,656,000  bu. 

Missouri. 
263,463,000  bu. 

Indiana. 
208,522,000  bu. 

Ohio. 
162,859,000  bu. 

Kentucky. 
126,859,000  bu. 

Kansas. 
121,097,000  bu. 

Tennessee. 
117,273,000  bu. 


For  the  construction  of  such  graphs,  squared  paper  is 
very  convenient.  Thus,  the  following  comparison  of  the 
wheat  crop  of  1917  in  the  five  leading  wheat  states  may 
be  shown  as  follows : 


Minn. 
N.  Dak. 
8.  Dak. 
Wash. 
Mont. 


DENOMINATE  NUMBERS 


161 


To  show  the  relative  changes  of  the  same  magnitude 
through  a  given  period  of  time,  both  the  straight  line  and 
the  broken  line  graphs  are  used.  Thus,  the  variation  of 
the  September  prices  of  hogs  ranging  over  a  period  of 
eight  years  may  be  represented  in  either  of  the  following 
ways: 


10 


8. 


•  II  I 

•  II  I 

•  II  I 

•  II  I 
I  I  I 


1910  '11  '12  '13  '14  '16  '18  '17 


13 


10 


1910  '11  '12  ,'13  '14  '15  '16  '17 


To  show  the  relation  of  a  part  to  the  whole,  a  circle  is 
more  often  used.  Thus,  the  following  graph  shows  how 
the  rent  was  apportioned  on  each  floor  of  three  stores 
during  a  recent  year.  It  is  a  portion  of  a  chart  given  in 
The  System,  December,  1917.  (See  page  162.) 

Pupils  should  be  encouraged  to  bring  in  graphs  found  in 
general  reading  and  interpret  them.  It  will  be  found 


162 


THE   TEACHING  OF  ARITHMETIC 


that  there  is  no  rigid  adherence  to  a  certain  form  for  a 
certain  type  of  comparison,  but  that  the  uses  shown  here 
are  general. 


Enough  work  in  the  construction  of  the  three  kinds 
shown  here  to  readily  interpret  those  found  in  general 
reading  should  be  done  if  time  permits. 


CHAPTER  XIII 
THE  PURPOSES  AND   NATURE   OF  PROBLEMS 

THE  great  advances  made  in  the  teaching  of  arithmetic 
during  recent  years  have  come  very  largely  through  an 
improvement  in  the  nature  of  the  problems.  This  im- 
provement has  followed  from  the  broader  conception  of 
the  purposes  of  a  problem  discussed  in  this  chapter,  for 
a  problem  is  no  longer  considered  a  mere  tool  to  train  the 
mind. 

In  the  lower  grades  the  problems  differ  greatly  hi  their 
nature  and  purpose  from  the  problems  of  the  upper  grades. 
For  that  reason  problems  will  be  discussed  under  two 
general  divisions :  the  problems  of  the  primary  grades, 
and  the  problems  of  the  upper  grammar  grades. 

THE  PROBLEMS  OF  THE  PRIMARY  GRADES 

While  the  child  in  the  first  four  or  five  grades  may 
get  several  things  as  by-products  of  his  study  of  a  problem, 
such  as  the  habit  of  looking  upon  the  quantitative  side 
of  life,  an  appreciation  of  the  value  of  arithmetic  in  doing 
the  world's  work,  something  of  a  social  insight  into  cer- 
tain phases  of  life,  etc.,  the  chief  purposes  of  the  problem 
during  these  years  are:  (1)  to  clarify  or  rationalize  the 

163 


164  THE  TEACHING  OF  ARITHMETIC 

facts  and  processes;  and  (2)  to  motivate  the  pure  drill 
work  in  the  fundamental  processes.  This  leads,  then,  to 
an  inquiry  into  the  nature  of  problems  that  will  do  this. 

CLARIFYING  A  PROCESS 

We  are  beginning  to  realize  that  the  child's  own  thought 
and  activity  should  dominate  classroom  instruction ;  that 
all  knowledge,  to  be  real,  must  be  based  upon  the  real 
experiences  of  the  individual  learner ;  and  that  arithmet- 
ical ideas  must  have  a  background  of  mental  images  which 
are  the  outgrowth  of  real  experiences,  in  order  to  exist. 
Hence,  all  facts  and  processes  should  be  presented  ob- 
jectively or  through  very  familiar  situations.  Through 
such  teaching  the  facts  have  a  real  background  of  vivid 
mental  images  and  the  pupil  is  able  to  use  them  in  the 
situations  that  arise  in  his  real  life.  The  chief  source  of 
failure  of  a  pupil  to  apply  his  arithmetical  facts  is  that 
they  have  been  put  before  him  as  mere  verbal  facts  with- 
out any  concrete  basis. 

When  a  problem,  however,  is  given  to  clarify  a  fact  or 
process,  it  must  be  made  from  real  objects  seen  and  handled 
by  the  class,  from  the  pictures  of  such  objects,  or  from 
experiences  that  are  very  familiar  to  the  pupils. 

It  is  through  developing  the  facts  of  addition,  multi- 
plication, etc.,  through  such  concrete  situations,  that  the 
pupil  gets  the  real  meaning  of  them  and  thus  knows  how 
to  use  them  in  new  situations  that  arise.  If  the  first 
problems  are  made  up  about  real  things  that  the  child 
sees,  then  next  about  pictures  that  call  up  real  objects, 
and  finally  about  things  that  he  easily  images,  he  grows 


PURPOSES  AND  NATURE  OF  PROBLEMS     165 

into  the  real  method  of  approaching  the  more  difficult 
problems  that  he  will  meet  later  in  the  course. 

MOTIVATING  DRILL  WORK 

Aside  from  rationalizing  a  fact  or  process,  the  problems 
given  during  the  early  years  of  a  child's  school  life  are 
given  very  largely  to  furnish  a  motive  for,  or  an  interest 
in,  adding,  subtracting,  multiplying,  and  dividing.  Such 
problems  should  require  little,  if  any,  thought  in  order 
'to  discover  the  process  to  be  used. 

To  motivate  the  drills,  however,  the  problems  will 
have  to  be  more  attractive  than  mere  drill  work  from 
abstract  numbers.  This  will  require  that  the  answer  is 
really  wanted  because  it  meets  some  personal  need  of 
the  pupil,  adds  in  some  way  to  his  pleasure,  or  satisfies 
his  curiosity. 

There  are  at  least  five  general  types  of  problems  that 
may  be  made  to  meet  these  conditions : 

(1)  Those  meeting  some  personal  question ;  as,  "If  you 
are  saving  money  to  buy  a  coaster  costing  $5.75  and  now 
have  $3.90,  how  much  more  will  you  need?" 

(2)  Those  having  to  do  with  some  home  affair;  as, 
"If  your  father  uses  12  tons  of  coal  each  winter,  costing 
$7.25  per  ton,  how  much  does  he  have  to  spend  for  coal  ?" 

(3)  Those  about  some  neighborhood  activity,  as  costs 
of  keeping  up  the  parks,  playgrounds,  schools,  etc. 

(4)  Those  that  appeal  to  his  civic  pride,  as  those  com- 
paring the  growth  of  his  town  or  state  with  some  other, 
or  those  about  some  industry  in  which  his  town  or  state 
excels. 


166  THE   TEACHING  OF  ARITHMETIC 

(5)  Problems  in  which  the  answer  simply  appeals  to 
his  curiosity,  as  the  number  of  things,  such  as  movie 
tickets,  baseball  bats,  etc.,  that  could  be  bought  for  the 
money  that  it  costs  to  run  the  government  for  a  day,  or 
with  the  wealth  of  some  of  the  well-known  men  of  wealth. 
These  furnish  larger  numbers  than  those  of  the  real 
problems  discussed  under  the  first  four  heads. 

THE  WORDING  OF  A  PROBLEM 

In  the  primary  grades,  as  has  just  been  pointed  out, 
the  child's  present  interest  is  the  fundamental  factor  to 
be  considered  in  selecting  or  making  problems.  The 
ultimate  use  of  the  subject  in  adult  activities  has  little, 
if  any,  place  in  such  a  selection  in  the  lower  grades.  That 
a  child  may  grow  into  the  true  method  of  solving  a  problem 
by  comparing  the  magnitudes  involved  and  thus  discov- 
ering whether  they  are  to  be  added,  subtracted,  multi- 
plied, or  divided,  the  early  problems  should  be  "story 
problems"  about  objects  present,  then  about  objects 
owned  by  the  children  or  familiar  to  them. 

Not  only  should  problems  be  about  familiar  objects, 
but  they  should  be  problems  that  the  child  might  very 
probably  ask.  Thus,  "Who  weighs  more,  you  or  I?" 
would  be  a  problem  that  a  child  would  be  much  more 
likely  to  ask  than,  "  How  much  do  you  and  I  both  weigh  ?  " 

In  the  early  years,  the  interest  is  determined  much 
more  by  the  wording  of  the  problem  than  by  the  numbers 
and  processes  involved.  Thus,  a  "story  problem"  such 
as,  "Mary  picked  8  pink  roses  this  morning.  She  is 
going  to  leave  5  at  home  for  her  mother  and  take  the  rest 


PURPOSES  AND  NATURE  OF  PROBLEMS     167 

to  her  teacher.  How  many  can  she  take  to  her  teacher  ?  " 
is  much  more  interesting  to  a  primary  pupil  than  the  same 
problem  stated  as,  "A  farmer  had  8  sheep  and  5  of  them 
died.  How  many  were  left?"  or  worse  yet,  "Eight 
horses  less  5  horses  are  how  many  horses?" 

It  would  be  much  more  interesting  and  do  much  more 
toward  developing  a  proper  method  of  solving  problems 
to  give  the  problem,  "If  you  had  put  9  little  rabbits  in 
a  pen  and,  when  you  went  to  feed  them,  found  that  there 
were  but  6,  how  many  were  gone  ?  "  than  to  give,  "  A  man 
had  9  cows  and  sold  6  of  them ;  how  many  were  left  ?  " 

It  is  much  better  to  say,  "Frank  got  50  Saturday  Eve- 
ning Posts  this  week  and  sold  all  but  4  of  them ;  how  many 
did  he  sell  ?"  than  to  say,  "A  farmer  had  50  acres  of  wheat 
and  he  harvested  all  but  4  acres ;  how  many  acres  has  he 
harvested?" 

In  making  problems,  then,  the  teacher  must  assure  her- 
self that  the  objects  are  very  real  to  the  children  and  the 
problem  is  so  woven  into  a  story  that  they  get  the  correct 
mental  picture  that  she  describes. 

THE  PROBLEMS  OF  THE  UPPER  GRAMMAR  GRADES 

There  are  some  who  seem  to  consider  that  the  develop- 
ment of  ability  to  compute  is  the  final  end  of  a  course  in 
arithmetic.  But  skill  in  computation  is  but  the  means 
to  an  end.  Those  who  would  try  to  meet  the  demands 
of  the  business  world  for  a  better  product  of  the  schools 
in  the  subject  of  arithmetic  by  merely  emphasizing  drill 
in  computation  are  following  a  course  about  as  inadequate 
for  the  purpose  of  developing  a  number  sense  or  a  mathe- 


168  THE   TEACHING  OF  ARITHMETIC 

matical  type  of  thought  as  a  training  that  confines  itself 
to  forming  the  letters  of  the  alphabet  would  be  as  a  final 
preparation  for  the  career  of  journalism. 
The  final  aims  in  a  course  in  arithmetic  are : 

(1)  To  develop  power  in  the  student  to  see  and  to  ex- 
press the  quantitative  relations  that  exist  among  the  mag- 
nitudes that  come  within  his  experience,  and  to  inter- 
pret the  numerical  expressions  of  such  relationships ; 

(2)  To  develop  in  the  student  the  habit  of  seeing  such 
relationships,  especially  those  vital  to  his  present  or  future 
welfare;  and 

(3)  To  give  the  student  a  social  insight  into  current 
business  and  industrial  practices  through  which  he  can 
interpret  references,  met  in  general  reading  and  in  con- 
versation, to  the  world's  activities. 

These  three  aims,  of  course,  come  through  the  problem 
side  of  arithmetic. 

POWER  TO  SEE  RELATIONS 

The  task  of  developing  power  to  see  and  to  use  quanti- 
tative relations  is  a  most  difficult  one.  When  we  think 
we  are  developing  such  power  we  are  often  merely  storing 
the  memory  with  rules,  forms,  and  processes.  If  the 
wording  is  changed,  pupils  are  often  at  sea  as  to  how  to 
proceed.  To  develop  power,  the  problems  and  the  word- 
ing must  be  so  varied  that  the  solution  has  to  depend 
upon  a  rational  analysis  of  conditions  and  not  be  a  mere 
act  of  memory.  This  power  can  never  come  through 
blind  adherence  to  rules  and  stereotyped  forms  of  solu- 
tion made  to  fit  artificial  types  of  problems. 


PURPOSES  AND  NATURE  OF  PROBLEMS     169 

The  first  requirement  of  a  problem  to  meet  this  pur- 
pose is  that  it  be  concrete  to  the  pupil.  Before  he  can 
analyze  it  and  thus  discover  the  steps  of  the  solution,  he 
must  have  a  very  clear  comprehension  of  the  facts  —  an 
accurate  mental  image  of  conditions. 

The  problems  must  not  only  be  very  concrete,  but  they 
must  suitably  task  the  pupil's  powers.  However  real  a 
problem  is,  a  pupil  is  much  more  interested  in  it  if  it  is 
difficult  enough  to  require  some  thought  in  order  to  deter- 
mine what  to  do.  A  pupil  soon  tires  of  long  lists  of  prob- 
lems all  solved  alike  where  he  gets  no  exercise  except  in 
computation.  Teachers  will  find  that  without  thought- 
provoking  problems  interest  will  soon  lag. 

While  problems  must  always  be  concrete,  they  need 
not  always  be  real.  In  the  development  of  this  power 
to  see  quantitative  relations,  the  hypothetical  problem 
still  has  its  place.  These  problems,  however,  must  never 
be  such  as  to  give  wrong  ideas  of  business  practices. 
One  does  not  have  to  resort  to  the  old  "hare  and  hound" 
type  of  problems  of  the  past  nor  develop  wrong  notions 
of  business  practice  to  get  problems  that  furnish  valuable 
training  of  power  to  see  and  express  relations.  The  data 
may  be  worth  while  and  the  problems  those  that  may 
arise  in  life.  For  example,  if  one  is  trying  to  develop  the 
power  to  see  and  to  interpret  the  relationships  expressed 
by  per  cent,  he  may  begin  with  a  certain  bit  of  data  and 
bring  in  all  the  various  relations  without  giving  any  wrong 
impression  of  business  practices  whatever.  For  example, 
let  it  be  given  that  it  cost  $38.45  to  manufacture  a  certain 
gasoline  engine,  and  the  cost  to  sell  it  amounted  to  $15.38 


170  THE    TEACHING   OF   ARITHMETIC 

more.    Now,  the  following  questions  may  be  asked  of  an 
advanced  class : 

1.  The  cost  to  sell  was  what  per  cent  of  the  cost  to 
manufacture  ? 

2.  The  cost  to  sell  was  what  per  cent  of  the  total  cost 
to  manufacture  and  sell  ? 

3.  At  what  must  it  be  sold  to  give  a  profit  of  25%  above 
the  cost  to  manufacture  and  sell  ? 

4.  At  what  must  it  be  sold  to  give  a  profit  of  25%  of 
the  selling  price  ? 

5.  At  what  price  would  it  have  to  be  listed  in  order  to 
allow  a  discount  of  40%  from  the  list  price  and  still  give 
a  profit  of  20%  of  the  total  cost  to  manufacture  and  sell  ? 

6.  At  what  price  must  it  be  listed  in  order  to  discount 
the  list  price  33£%  and  10%  and  still  make  a  profit  of 
20%  of  the  actual  selling  price  ? 

And  thus  one  may  go  on  making  up  a  lot  of  questions 
of  increasing  difficulty  to  meet  the  maturity  of  the  pupil's 
powers,  that  have  great  value  in  bringing  out  the  rela- 
tions expressed  by  per  cent.  While  one  hears  a  great 
amount  of  criticism  of  any  but  "real"  problems,  it  ought 
to  be  clear  to  any  teacher  that  such  criticism  is  not  based 
upon  a  careful  consideration  of  the  aims  to  be  attained. 

THE  HABIT  OF  SEEING  RELATIONS 

One  of  the  greatest  sources  of  waste  in  so  much  of  our 
teaching  is  that  pupils  are  not  so  taught  that  they  can 
apply  the  knowledge  gained  in  the  classroom  to  the  work 
found  outside  of  the  classroom.  There  should  be  a  con- 
stant utilization  of  outside  experiences  to  clarify  and 


PURPOSES  AND  NATURE  OF  PROBLEMS     171 

rationalize  the  work  of  the  schoolroom,  and  likewise  the 
facts  learned  in  the  schoolroom  should  constantly  be 
used  to  interpret  conditions  met  without  the  school- 
room. Every  subject  in  the  textbook  should  be  supple- 
mented by  "community  problems"  made  from  data 
gathered  by  the  teacher  and  pupils  from  the  immediate 
community  in  which  they  live.  It  is  only  through  mak- 
ing such  problems  that  pupils  develop  the  habit  of  look- 
ing upon  the  quantitative  side  of  life.  Too  often  the 
problems  of  the  text  mean  no  more  to  them  than  mere 
assignments  in  order  to  see  if  they  can  get  the  answers. 
However,  with  the  improved  type  of  problems  that  are 
beginning  to  find  their  way  into  our  newer  texts,  a 
teacher  can  do  a  great  deal  toward  developing  proper 
habits  by  making  the  right  use  of  them.  But  the  pupil 
must  be  taught  to  picture  the  situation  and  to  ask  him- 
self if  the  conditions  of  the  problem  meet  local  condi- 
tions, whether  the  answer  is  a  reasonable  one,  etc. 

Problems  of  thrift  are  among  the  most  useful  in  the 
development  of  the  habit  of  seeing  those  quantitative 
relations  vital  to  our  welfare.  Thus,  such  questions 
may  be :  How  much  is  saved  by  buying  enough  potatoes 
for  the  winter  direct  from  the  farm  at  digging  time  in- 
stead of  buying  them  by  the  peck  from  the  grocer  during 
the  winter?  How  much  is  saved  by  buying  canned  goods 
by  the  case  instead  of  by  the  can?  How  much  is  saved 
by  buying  material  and  making  your  own  dress  instead 
of  buying  it  ready  made?  Will  it  pay  better  to  rent  a 
house  or  to  own  one?  Will  it  pay  in  your  town  to  build 
a  house  for  rent  ?  About  how  much  could  you  expect  to 


172  THE   TEACHING  OF  ARITHMETIC 

make  net  upon  your  investment?  Has  the  price  upon 
vacant  lots  in  your  town  gone  up  so  that  the  one  holding 
them  for  a  certain  time  made  or  lost?  In  fact,  there  is 
no  end  to  good  problems  that  may  be  made  from  the 
material  found  in  any  community. 

PROBLEMS  THAT  CONTRIBUTE  TO  SOCIAL  INSIGHT 

As  the  pupil's  social  world  expands,  he  is  carried  from 
problems  dealing  with  his  own  personal  affairs  and  local 
situations  to  those  of  more  general  interest.  However, 
the  teacher  must  have  a  much  broader  concept  of  such 
problems  than  that  they  are  to  meet  the  demands  of  the 
commercial  and  industrial  world.  They  should  be  prob- 
lems that  contribute  to  the  pupil's  interpretation  of  the 
social  world  in  which  he  lives  —  that  contribute  to  "  a 
broadly  socialized  and  modern  culture,"  not  a  mere  voca- 
tional training.  That  is,  a  study  of  arithmetic  should 
enable  one  to  interpret  references  met  in  general  reading 
and  in  social  intercourse  to  all  the  common  affairs  of  life, 
as  taxes,  insurance,  interest,  stocks,  bonds,  notes,  mort- 
gages, and  other  business  forms,  and  to  any  references  to 
measurements  and  relations,  to  the  statistical  relations 
represented  by  graphs,  and  to  any  other  phases  of  the 
subject  through  which  he  can  better  interpret  the  quanti- 
tative side  of  his  environment. 

To  do  this,  the  problems  must  be  real,  true  to  present- 
day  conditions,  and  not  fictitious  problems  about  real 
things.  But  however  "real"  they  are,  they  must  be  full 
of  human  interest  to  the  child.  The  following  problem 
given  in  a  recent  textbook  for  third  grade  pupils  may  be 


PURPOSSS  AND  NATURE  OF  PROBLEMS     173 

real,  but  it  lacks  any  element  that  makes  it  interesting  to 
a  child  of  nine  years.  The  problem  is :  "  If  a  200  pound 
sack  of  fertilizer  contains  8  pounds  of  nitrogen,  10  pounds 
of  potash,  and  16  pounds  of  phosphoric  acid,  how  many 
pounds  of  these  plant  foods  are  there  in  a  sack?" 

Simply  because  a  problem  is  "real"  to  one  in  some 
specialized  vocation,  it  does  not  follow  that  it  necessarily 
contributes  to  the  social  insight  of  the  pupil.  Unless  the 
problem  is  one  likely  to  come  up  in  interpreting  some 
phase  of  life  which  is  of  interest  to  the  student  either  at 
the  time  or  at  a  probable  future  time,  it  is  useless  from  this 
standpoint. 

It  is  not  to  be  understood  that  a  problem  meets  but 
one  of  the  three  aims  of  a  problem  at  a  time.  A  good 
problem  may,  and  often  does,  meet  all  three  at  once. 
The  problems  to  meet  the  last  condition  discussed  will 
in  general  meet  this  one.  The  necessary  conditions  are 
that  the  problems  answer  some  question  arising  from 
some  social  issue,  and  that  the  data  be  real  and  of  enough 
interest  to  warrant  their  use.  Such  problems  are  as  varied 
as  the  activities  of  life.  Those  of  greatest  value  are  the 
ones  having  to  do  with  industrial  and  commercial  terms 
and  practices,  but  they  are  not  all  included  in  these  topics. 
They  may  relate  to  all  the  various  productions  needed 
for  the  world's  comfort  —  the  amount,  the  value,  the 
means  and  cost  of  the  distribution  of  them,  and  so  forth. 


CHAPTER  XIV 


DIFFICULTIES  IN  TEACHING 

IT  is  the  experience  of  teachers  that  to  teach  a  pupil 
how  to  interpret  and  solve  a  problem  is  a  much  more 
difficult  task  than  to  develop  rules  and  train  for  skill  in 
computation.  And  it  is  not  strange  that  this  is  so.  The 
formal  side  —  the  methods  of  computation  —  is  merely  a 
machine  which  the  pupil  learns  to  operate,  and  it  works 
the  same  upon  all  occasions  and  under  all  conditions. 
But  the  solution  of  a  problem  is  an  application  of  judg- 
ment. The  problem  must  be  analyzed,  the  conditions 
must  be  carefully  studied,  the  magnitudes  wanted  must 
be  compared  with  those  given,  and  an  act  of  judgment 
tells  what  processes  are  to  be  used.  The  solution  of  a 
problem  cannot  be  done  by  rote,  as  the  process  of  addi- 
tion or  multiplication  can  be  done.  The  solution  can  be 
discovered  only  through  a  contribution  from  the  experi- 
ences of  the  individual  and  an  act  of  judgment.  And, 
hence,  if  the  necessary  experiences  are  lacking,  the  prob- 
lem cannot  be  solved. 

It  is  obvious,  then,  that  no  rule  can  be  laid  down  that 
will  teach  a  pupil  how  to  solve  a  problem ;  but  a  discus- 
sion of  the  sources  of  failure  may  prove  profitable  to  the 
teacher. 

174 


ANALYSIS  AND  SOLUTION  OF   PROBLEMS     175 


When  a  problem  is  difficult  to  any  large  number  of  a 
class,  the  teacher  cannot  lay  the  blame  to  the  dullness  of 
the  children.  A  large  per  cent  of  our  pupils  are  capable, 
normal  children.  Then  we  must  look  for  the  trouble 
either  in  the  material  we  are  using  or  in  our  method  of 
presenting  it.  If  we  examine  the  material  found  in  our 
textbooks,  we  shall  find  much  that  may  be  very  justly 
condemned.  While  there  has  been  a  very  marked  im- 
provement in  our  textbooks  in  recent  years,  many  of  the 
problems  still  have  but  little  meaning  and  but  little  appeal 
to  children.  Many  of  the  problems  designed  for  the  pri- 
mary grades  are  taken  from  the  adult's  world  and  not  the 
child's  world.  They  are  about  business,  manufacturing, 
transportation,  and  various  industries  with  which  chil- 
dren have  no  vital  contact,  instead  of  problems  about 
the  child's  play,  his  daily  activities  about  the  home,  or 
his  constructive  activities.  In  so  far  as  textbooks  have 
poorly  chosen  their  problem  material,  they  have  contrib- 
uted directly  to  the  trouble  in  question  and  hence  are 
at  fault.  But  the  material  in  the  textbook  is  not  the  only 
available  material.  A  resourceful,  wide-awake  teacher 
who  understands  the  child's  interests  and  needs  and  who 
understands  the  fundamental  principles  of  teaching  will 
be  able  to  obtain  admirable  results  in  spite  of  a  poor 
book. 

A  FUNDAMENTAL  PRINCIPLE  OF  TEACHING 

At  the  very  root  of  all  successful  teaching  lies  the 
fundamental  principle  that  all  knowledge  possessed  by 


176  THE   TEACHING  OF  ARITHMETIC 

the  individual  to  be  real  and  permanent  must  be  grounded 
upon  and  developed  out  of  his  real  experiences. 

And  thus  all  arithmetical  ideas  must  have  a  back- 
ground of  mental  images  in  order  to  exist.  These  mental 
images  in  turn  must  be  the  outgrowths  of  some  form  of 
experience.  Henca,  the  real  ideas  which  constitute  the 
child's  arithmetical  knowledge  must  be  developed  through 
the  individual  experiences  of  the  learner  which  have  been 
provided  for  either  in  or  out  of  school.  The  failure  to 
provide  these  clear  mental  images,  which  the  statement 
of  a  problem  should  call  up  in  the  child's  mind,  is  perhaps 
the  chief  source  of  all  failure  to  solve  a  problem. 

CONCRETE  PROBLEMS 

Another  way  of  stating  this  source  of  failure  to  solve 
a  problem  is  to  say  that  the  problem  was  not  concrete  to 
the  pupil ;  that  is,  he  was  not  able  to  form  a  clear  mental 
picture  of  the  situation  described.  Unless  such  a  picture 
can  be  formed,  the  pupil  is  unable  to  make  a  comparison 
of  what  is  wanted  with  what  is  given  in  order  to  form 
a  judgment  of  what  to  do  to  obtain  the  desired  result. 

When  a  pupil  fails  to  form  such  an  image,  it  may  be 
that  he  has  not  read  the  problem  carefully.  Or  he  may 
have  read  it  carefully  and  yet,  through  lack  of  experi- 
ence in  the  situation  described,  he  may  not  be  able  to  form 
a  picture  of  the  situation.  Recently,  a  little  girl  in  the 
fourth  grade  told  me  that  her  pencil  cost  five  cents.  I 
asked  her  how  much  two  such  pencils  would  cost.  She 
told  me  at  once,  of  course,  and  then  she  reminded  me  that 
she  was  in  the  fourth  grade  and  that  such  problems  were 


ANALYSIS  AND  SOLUTION  OF  PROBLEMS    177 

too  easy  for  her.  So  I  said,  "Well,  try  this  one:  If  a 
meter  of  silk  costs  five  francs,  how  much  will  two  meters 
cost?"  She  could  not  tell  me.  She  said,  "I  don't  know 
what  a  meter  is  and  I  don't  know  what  francs  are,  so  of 
course  I  can't  solve  it." 

Some  months  ago  I  was  trying  to  explain  to  a  young 
teacher  that  the  quotient  of  the  length  of  a  row,  divided 
by  the  distance  between  plants,  did  not  give  the  number 
of  plants  in  the  row;  but  that  the  quotient  was  merely 
the  number  of  divisions  into  which  the  plants  divided 
the  row,  and  hence  there  must  be  one  more  plant  than 
division.  After  I  had  exhausted  my  stock  of  word  pic- 
tures, I  took  a  pencil  and  began  a  diagram,  when  the 
teacher  to  whom  I  was  explaining  the  problem  exclaimed : 
"Oh,  I  see  it.  It  is  just  like  hanging  handkerchiefs  on  a 
clothesline.  It  takes  one  more  clothespin  than  there 
are  handkerchiefs."  I  had  finally  called  up  some  expe- 
rience in  her  own  We  through  which  the  problem  became 
concrete  to  her. 

Sometimes  the  largeness  of  the  magnitudes  involved 
takes  away  the  concreteness  of  the  situation,  and  the 
mind  sees  but  the  empty  figures  without  any  concrete 
basis  back  of  them.  I  recently  saw  a  student  attempt 
to  find  the  capacity  of  a  bin,  having  been  given  the  di- 
mensions and  also  the  statement  that  one  cubic  foot  was 
equal  to  .8  of  a  bushel.  After  correctly  finding  the  num- 
ber of  cubic  feet  in  the  bin,  he  divided  by  .8  to  find  the 
number  of  bushels.  But  the  teacher  said,  "  What  part 
of  a  bushel  does  one  cubic  foot  equal?  Then  what  does 
2  cubic  feet  equal?  What  does  3  cubic  feet  equal?" 


178  THE   TEACHING   OF   ARITHMETIC 

These  were  correctly  answered ;  then  the  pupil  saw  clearly 
that  he  should  multiply  .8  of  a  bushel  by  the  number 
of  cubic  feet  in  the  volume  in  order  to  get  the  capacity  in 
bushels. 

The  only  true  aid  that  the  teacher  can  give  is  to  help 
the  pupil  translate  the  problem  into  his  own  experiences. 
Power  to  do  this  cannot  be  given  by  any  book  upon 
"method,"  but  depends  upon  the  personality,  versatility, 
and  experiences  of  the  teacher.  Teachers  need  to  get  as 
wide  a  contact  as  possible  with  that  side  of  life  to  which 
arithmetic  is  applied  in  order  to  give  practical  and  con- 
crete applications  to  every  topic  of  the  subject.  It  is 
only  through  such  knowledge  that  the  most  efficient  work 
can  be  done  in  the  application  side  of  the  subject. 

BOOK  PROBLEMS  vs.  LOCAL  PROBLEMS 

The  problems  of  any  textbook  must  of  necessity  be 
designed  for  pupils  of  all  communities.  They  must  draw 
upon  the  experiences  of  all  children  alike.  They  cannot 
take  into  account  the  different  environments  of  children 
of  different  localities.  Evidently,  then,  all  problems 
cannot  be  equally  concrete  to  all  children.  Hence,  it 
follows  that  the  textbook  should  be  supplemented  by 
problems  taken  from  the  child's  own  local  environment. 
These  problems  are  not  only  more  concrete  to  him  but, 
if  dealing  with  affairs  in  which  he  is  interested,  they  fur- 
nish a  much  stronger  motive  for  learning  arithmetic  than 
do  those  of  the  book.  These  local  problems  may  deal 
with  personal  or  family  purchases,  with  the  child's  con- 
structive activities,  with  local  stores  and  markets,  econ- 


ANALYSIS   AND  SOLUTION  OF   PROBLEMS     179 

omy  in  buying  in  quantities,  local  improvements,  farm 
crops,  etc.  The  data  for  these  problems  should  be  gath- 
ered by  both  teacher  and  pupils,  and  much  of  the  problem 
making  should  be  done  by  the  pupils  themselves.  In 
this  way  arithmetic  provides  a  much  better  equipment 
for  life  than  can  be  obtained  from  the  textbook  alone, 
for  the  problems  encountered  in  real  life  are  not  in  general 
formulated  for  us;  but  from  known  data  the  individual 
must  formulate  his  own  problem.  In  so  far  as  possible, 
the  school  should  provide  conditions  for  incidental  prob- 
lem work  in  the  various  school  activities.  These  prob- 
lems, however,  must  be  "unavoidable"  problems  that 
arise  and  must  be  solved.  In  other  words,  they  should 
not  be  problems  that  are  assigned  merely  in  order  to  have 
a  problem  that  will  give  drill  in  certain  processes,  but  the 
problems  must  be  those  in  which  it  is  necessary  to  know 
the  answer  and  not  problems  giving  results  for  which  there 
is  no  need  or  interest.  The  making  of  "number  stories" 
by  the  children  is  a  beginning  in  this  important  phase  of 
the  problem  work. 

OTHER  SOURCES  OF  FAILURE 

While  no  doubt  the  chief  source  of  failure  to  solve 
a  problem  is  the  lack  of  a  clear-cut  mental  picture  of  the 
situation  described,  there  are  other  causes  that  should 
be  discussed.  If  a  teacher  will  list  the  reasons  why  a 
pupil  fails  to  get  the  correct  answers  to  assigned  problems, 
she  will  probably  find  that  the  failures  will  fall  under  the 
following  general  heads : 

1.  The  problem  was  not  concrete. 


180  THE   TEACHING  OF   ARITHMETIC 

2.  The  computation  was  inaccurate. 

3.  Approximations  were  made  before  the  computation 
was  completed. 

4.  The  meaning  of  the  fundamental  processes  was  not 
understood. 

5.  The  author  of  the  problem  assumed  facts  that  were 
not  known  by  the  pupils. 

6.  The  problem  was  too  difficult ;  that  is,  the  relations 
existing  among  the  data  were  too  complex.    In  other 
words,  the  pupil  lacked  the  mental  power  to  reason  out 
what  processes  to  use. 

INACCURATE  COMPUTATION 

The  lack  of  concreteness  has  already  been  discussed. 
Perhaps  errors  in  computation  are  as  frequent  a  cause 
of  wrong  results  as  lack  of  concreteness.  And  yet  this 
source  of  error  may  be  practically  eliminated.  From  the 
very  first  step  in  the  teaching  of  the  written  processes, 
the  pupil  should  be  taught  to  check  his  work.  He  should 
feel  from  the  first  that  when  a  computation  has  been 
performed  but  once  it  is  but  half  done.  It  must  be  re- 
viewed or  checked  in  some  way  to  insure  its  accuracy. 
This  is  done  in  every  vocation.  However  small  the 
amount,  a  clerk  always  checks  every  sale-slip.  To  make 
one  error  out  of  a  thousand,  a  clerk  would  lose  his  posi- 
tion. Yet  a  pupil  is  allowed  to  hand  in  work  without 
checking  it  and,  if  he  gets  but  two  out  of  ten  answers 
wrong,  he  is  given  "80%"  or  "good."  Some  schools 
are  taking  as  their  motto :  "  One  hundred  per  cent  accu- 
racy in  all  computation  is  our  aim."  This  does  not  mean 


ANALYSIS  AND  SOLUTION  OF   PROBLEMS     181 

that  pupils  are  never  expected  to  make  a  mistake  but 
that,  just  as  in  business  life,  their  computations  are  to 
be  checked  until  they  know  when  work  is  turned  in  that 
it  is  100%  accurate. 

But  this  habit  cannot  be  developed  as  long  as  teachers 
call  the  work  "good"  when  two  out  of  every  ten  answers 
are  wrong.  Higher  standards  will  have  to  be  required 
in  order  to  furnish  a  motive  for  checking  the  computation. 

ERROR  IN  APPROXIMATING  RESULTS 

Even  if  all  computation  has  been  carefully  checked, 
a  pupil  may  fail  through  making  an  approximation  before 
the  computation  is  completed.  Pupils  should  be  taught 
to  see  clearly  just  what  effect  an  approximation  during 
the  computation  is  going  to  have  on  the  final  result. 
Recently  a  pupil  was  "just  sure  the  answer  book  is  wrong" 
to  the  following  problem  :  "  How  many  cubic  feet  of  water 
can  a  V-shaped  gutter  discharge  (flowing  full)  in  a  day  if 
it  is  8  inches  deep  and  16  inches  wide  at  the  top,  and  the 
water  flows  a  foot  a  second?" 

He  had  reasoned  correctly  and  his  computation  was 
correct  as  far  as  he  had  carried  it.  He  saw  that  the 
problem  was  to  find  the  volume  of  a  triangular  prism 
whose  base  was  the  cross-section  of  the  gutter  and  whose 
height  was  60X60X24  feet.  His  error  came  from  first 
reducing  the  cross-sectional  area  (64  square  inches)  to 
a  decimal  part  of  a  square  foot  "true  to  thousandths" 
before  multiplying  by  86,400.  He  did  not  realize,  how- 
ever, that  the  error  in  approximating  the  third  decimal 
was  multiplied  86,400  times  in  his  answer. 


182  THE   TEACHING  OF  ARITHMETIC 

Pupils  should  be  taught  that,  when  a  solution  consists 
of  multiplications  and  divisions  only,  all  work  should 
first  be  indicated,  then  all  common  factors  canceled  from 
both  dividend  and  divisor,  and  all  multiplications  in  the 
dividend  performed  before  any  of  the  division  is  done. 
Had  this  been  done,  the  problem  given  above  could  have 
been  solved  without  any  use  of  a  pencil  except  in  the 
cancellation. 

FORECASTING  RESULTS 

After  a  pupil  has  "thought  out"  what  to  do  in  the  solu- 
tion of  a  problem,  he  should  be  required  to  estimate  the 
result.  Such  an  estimate  serves  as  a  check  upon  absurd 
errors  in  computation,  such  as  misplacing  the  decimal 
point,  failing  to  multiply  by  all  factors,  etc. 

But  the  habit  of  making  close  approximations  has  a 
very  much  more  important  use  than  merely  to  help  check 
up  the  work  of  computation.  In  practical  life,  a  mere 
approximation  of  a  result  often  suffices.  So,  if  one  is 
trained  to  make  rather  close  approximations  without  a 
pencil,  he  is  getting  a  training  in  a  very  practical  phase 
of  the  subject  —  power  to  see  the  world  about  him  from 
the  quantitative  standpoint. 

MEANING  OF  THE  PROCESSES  NOT  KNOWN 

It  sometimes  occurs  that  a  pupil  does  not  know  what 
process  to  apply  because  he  does  not  know  what  the  pro- 
cesses mean.  I  once  asked  a  little  girl  in  the  third  grade 
the  question,  "Five  threes  make  how  many?"  She 
looked  at  me  in  bewilderment  and  then  asked,  "Do  you 


ANALYSIS   AND  SOLUTION  OF   PROBLEMS     183 

mean  and  or  times  or  less  ?  "  She  would  no  doubt  have  had 
trouble  in  applying  her  facts  to  the  solution  of  problems. 
,Not  understanding  the  meaning  of  a  process,  or  not 
seeing  just  what  it  means  when  applied  to  a  given  situa- 
tion, often  leads  to  error.  In  the  problem,  "How  many 
plants  8  inches  apart  can  be  set  in  a  row  which  is  to  be 
20  feet  long  from  the  first  to  last  plant  ?  "  the  pupil  usually 
divides  240  inches  by  8  inches  and  answers  "30  plants," 
not  seeing  what  the  division  actually  means  —  that  it 
is  finding  how  many  times  the  row  will  contain  an  8-inch 
measure  and  that,  from  the  number  of  times  it  is  contained, 
the  number  of  plants  must  be  found  by  visualizing  the 
act  of  placing  a  plant  at  each  end  of  the  measure  the  first 
time  it  is  applied  and  then  one  plant  for  each  new  space 
measured  off. 

In  the  problem,  "  From  a  sheet  of  Manila  board  22  inches 
by  28  inches,  how  many  cards  8  inches  by  11  inches  can 
be  cut?"  the  pupil  often  answers  "7."  He  jumps  at 
the  conclusion  that  it  is  one  area  divided  by  the  other  and 
does  not  see  that  this  is  not  the  same  as  actually  laying  off 
the  sheet  into  cards,  in  which  case  he  would  find  that  he 
could  get  but  six. 

NECESSARY  FACTS  NOT  KNOWN 

Sometimes  a  pupil  has  the  correct  picture  of  the  situa- 
tion described  in  a  problem,  but  lacks  a  knowledge  of 
the  facts  upon  which  the  solution  depends.  This  failure 
occurs  most  frequently  in  the  problems  of  mensuration 
where  the  pupil  does  not  know  how  to  find  certain  areas 
or  volumes.  Thus,  I  recently  saw  a  pupil  attempt  the 


184  THE   TEACHING  OF  ARITHMETIC 

solution  of  the  following  problem:  "The  cross-section  of 
a  railroad  tunnel  132  yards  long  is  in  the  form  of  a  16-ft. 
square  surmounted  by  a  semicircle.  Find  the  number 
of  cubic  yards  of  earth  removed  in  its  excavation."  The 
pupil  stepped  to  the  blackboard  and  sketched  accurately 
the  picture.  She  stated  that  the  problem  was  to  find  the 
volume  of  a  prism  and  half  that  of  a  cylinder  and  add 
them.  She  then  failed  in  the  solution  because  she  tried 
to  find  the  volume  by  multiplying  the  perimeter  of  the 
cross-section  by  the  length  of  the  tunnel.  In  a  case  of 
this  kind,  there  is  nothing  to  do  but  re-teach  these  funda- 
mental rules.  It  is  not  sufficient  to  ask  the  pupil  to 
"review  the  rules  for  to-morrow,"  but  the  rules  should 
be  again  presented  as  concretely  as  possible.  Pure 
memorization  of  facts  without  the  mental  image  of 
personal  experiences  as  a  basis  is  almost  valueless.  If 
all  rules  of  mensuration  and  all  other  facts  of  arithmetic 
are  presented  objectively,  they  are  much  more  easily 
retained  than  when  learned  by  rote.  The  vivid  mental 
image  growing  out  of  an  experience  is  much  more  lasting 
than  a  fact  memorized  through  the  repetition  of  meaning- 
less language. 

COMPLEXITY  OP  RELATIONS 

Sometimes  a  pupil  cannot  solve  a  given  problem  be- 
cause the  relations  existing  among  the  data  are  too  com- 
plex. That  is,  failure  may  be  due  to  lack  of  sufficient 
power  to  reason.  However,  pupils  often  seem  to  fail 
from  lack  of  reasoning  power  when  the  real  trouble  is 
that  they  have  made  no  serious  attempt  to  analyze  the 


ANALYSIS  AND  SOLUTION  OF  PROBLEMS    185 

situation  and  thus  to  discover  the  essential  relations  that 
exist.  They  depend  too  largely  upon  some  word  or  words 
in  the  description  of  the  situation,  and  then,  through 
memory  alone,  associate  the  problem  with  some  one  en- 
countered in  the  past  experience  for  their  "cue"  as  to 
what  to  do,  rather  than  make  any  serious  analysis  of  the 
conditions  described. 

When  the  answer  to  a  problem  is  known,  rather  than 
reason  out  the  solution,  the  pupil  is  very  apt  to  try  to 
juggle  the  figures  in  some  way  that  will  give  the  result. 
Recently  I  saw  a  pupil  attempt  the  following  problem : 
"If  the  interest  of  $500  for  2^  years  is  $50,  what  is  the 
rate?"  She  knew  that  the  answer  was  4%.  Her  solu- 
tion was  as  follows : 

2£x$50=$125;    $500 -J- $125  =  4% 

In  most  schoolrooms,  the  teaching  of  a  systematic 
method  of  attack  in  the  solution  of  a  problem  is  greatly 
neglected.  The  assignment  of  "the  next  ten  problems" 
with  no  thought  given  as  to  their  fitness  for  the  class  is 
entirely  too  common  a  practice.  The  recitation  should 
not  be  a  place  in  which  the  teacher  is  to  find  out  whether 
assignments  have  been  finished,  but  it  should  be  a  work- 
shop in  which  the  pupil  is  taught  to  translate  the  arithmet- 
ical question  into  terms  of  his  own  experiences. 

Home  work  should  be  given  but  sparingly  and  should 
consist  chiefly  in  performing  the  mechanical  computa- 
tions involved  after  the  "what  to  do"  has  been  discussed 
in  the  classroom.  When  pupils  are  assigned  home  work 
that  they  do  not  understand  and  have  to  depend  upon 


186  THE    TEACHING  OF   ARITHMETIC 

help  given  by  the  parent,  there  is  no  uniformity  of  the 
instruction  of  the  various  members  of  the  class;  and,  in 
general,  the  home  instruction  differs  from  that  given  by 
the  teacher.  Hence,  confusion  arises.  But  worse  than 
that  is  the  fact  that  much  of  the  home  work  brought  in 
by  the  pupil  is  not  his  own  but  that  of  his  parents,  and 
hence  the  teacher,  knowing  that  the  work  is  brought  in, 
does  not  realize  the  difficulties  that  the  child  has  encoun- 
tered or  that  he  does  not  fully  understand  the  work. 

The  classroom  period  should  be  given  up  to  instruction 
—  to  real  teaching  —  and  to  rapid  motivated  drills. 
The  working  out  of  mechanical  details,  as  computing  re- 
sults, etc.,  should  be  the  only  part  left  for  home  work,  in 
most  grades  at  least. 

It  is  a  common  experience  of  teachers  that  pupils  who 
can  solve  problems  when  stated  in  the  familiar  terms  of 
the  textbook  in  certain  stereotyped  forms  of  expression 
fail  to  solve  the  same  problems  when  stated  in  language 
less  common  or  when  met  in  actual  life.  To  overcome 
this  defect  demands  a  more  varied  use  of  language  in  the 
description  of  a  situation  as  well  as  a  wider  range  of  ma- 
terial used. 

To  develop  a  systematic  method  of  attack  in  solving 
a  problem,  the  pupil  must  be  led  through  simple  types 
of  one-step,  two-step,  and  three-step  (or  more)  problems, 
all  stated  in  various  ways  and  descriptive  of  a  wide  range 
of  material,  all  within  the  range  of  his  own  experiences 
and  interests.  He  must  state  the  problem  in  his  own 
words,  pointing  out  just  what  is  wanted  and  what  is  given 
in  the  problem  that  will  help  find  this.  Then,  from  a 


ANALYSIS   AND   SOLUTION  OF   PROBLEMS     187 

clear  mental  image  of  the  things  involved,  he  will  see 
the  essential  relations  that  exist  and  will  know  what 
to  do.  The  numbers  involved  in  any  new  problem  should 
be  so  small  and  the  computation  so  easy  as  not  to  retard 
thinking. 

THE  SOLUTION  OF  PROBLEMS 

The  solution  of  a  problem  requires  an  act  of  judgment 
and  cannot  be  made  merely  the  application  of  a  rule  or 
formula.  Hence,  there  is  as  large  a  variety  of  solutions  as 
there  are  variations  in  the  nature  of  problems. 

After  reasoning  out  what  to  do,  the  computation  should 
be  made  as  simple  as  possible.  When  there  are  several 
steps  in  the  solution,  it  is  best  to  reason  entirely  through 
to  the  end  before  performing  the  computation.  Some- 
times one  step  will  balance  or  cancel  a  later  one  and  thus 
work  is  saved.  For  example,  to  find  the  cost  of  18,760 
Ib.  of  hay  at  $18  per  ton,  it  is  seen  that  to  divide  18,760 
by  2000  gives  the  number  of  tons,  which  is  9.38.  Then 
9.38  X $18  gives  the  cost  of  all.  But  observing  the 
processes  to  be  performed,  we  see  that  by  multiplying 
18,760  by  9  and  pointing  off  three  decimal  places  the  wprk 
can  all  be  done  without  a  pencil.  If  the  work  is  to  be 
written  down,  it  should  be  done  in  such  a  way  that  the 
pupil  does  not  think  that  he  is  multiplying  pounds  by 
something  else  and  getting  dollars.  The  solution  might 
be  written  out  logically  as  follows : 

9 

I|^X$Z8  =  $170.840 
2000 

1000 


188  THE   TEACHING  OF  ARITHMETIC 

In  finding  18,760  X  $9,  the  logical  multiplier  is  18,760, 
but  the  actual  one  is  9.  The  denomination  of  the  answer 
depends  upon  the  denomination  of  the  number  multi- 
plied, of  course;  but  the  real  process  is  that  of  finding 
the  product  of  the  two  abstract  factors,  18,760  and  9, 
either  of  which  may  be  used  as  the  multiplier. 

When  several  multiplications  and  divisions  occur, 
all  work  should  be  written  down,  like  factors  canceled, 
and  the  multiplication  performed  before  dividing.  Most 
problems  of  mensuration  are  of  this  type.  Thus  to  find 
the  capacity  of  a  bin  15  feet  long,  8^  feet  wide,  and  filled 
to  a  depth  of  6  feet,  using  .8  bu.  per  cubic  foot,  the 
solution  is  : 


2  0 

After  canceling  like  factors  the  multiplication  can  be 
performed  without  a  pencil. 

To  find  the  capacity  of  a  tank  27  inches  by  42  inches 
by  22  inches,  using  231  cubic  inches  per  gallon,  the  solu- 
tion is  as  follows  : 


Again  all  computation  is  done  without  a  pencil.  In- 
stead of  writing  231,  its  factors  were  written  to  facilitate 
cancellation. 

To  discount  a  bill  of  $134.50  33£%  and  10%  work  is 


ANALYSIS  AND  SOLUTION  OF   PROBLEMS     189 

saved  by  discounting  by  10%  first,  for  134,50  will  not 
contain  3;    but   after  taking  off  10%  of   any 
number  the   remainder  will   always  contain  3.        fq    ^ 
Hence,  the  solution  might  be  as  given  in  the  ' 


margin.  4Q  35 

The  same   problem,  however,  can  be  solved 

without  a  pencil  by  observing  that,  after  33^% 

and  10%  have  been  deducted,  but  60%  remains.     And 

60%  of  $134.50  =  $80.70. 
No  fixed  form  of  solution  should  be  required.    The 

pupil  should  reason  out  what  to  do  and  look  to  see  if  any 

required  process  will  cancel  any  other. 

ERRORS  IN  LABELING  STEPS  IN  THE  SOLUTION 

It  is  a  common  practice  to  require  the  pupil  to  label 
each  step.  This  invariably  leads  to  erroneous  reasoning. 
This  is  clearly  shown  in  the  usual  solution  to  the  follow- 
ing type  of  problems:  "Allowing  7  pk.  of  seed  wheat 
to  the  acre,  how  many  bushels  are  needed  for  a  field  35  rd. 
by  42  rd.?"  In  many  schoolrooms  you  will  see  the  solu- 
tion shown  in  the  margin.  It  will 
be  seen  that  every  principle  of  multi-  35  rd. 

plication  and  division  has  been  violated.  42  rd. 

Rods   cannot   be   multiplied   by   rods.  70 

Square  rods  divided  by  160  does  not          140. 
give    acres.    Acres    multiplied    by    7   160)1470  sq.  rd. 
does  not  give  pecks ;  and  pecks  divided  9  ^  A. 

by  4  does  not  give  bushels.     None  of  7 

the  steps  should  have  been  labeled  in  4)64  ^  pk. 

such  a  solution.    The  following  solution  16  -fa  bu. 


190  THE   TEACHING  OF   ARITHMETIC 

would  not  only  have  been  correct,  but  it  would  have  been 
much  more  economical : 

7     21 

30X42^7,          1029  , 
-— — —  X  -.  bu.  =  — —  bu.  =16^  bu. 
Z00        4  64 

32         2 

It  is  not  the  type  of  solution,  however,  that  is  impor- 
tant. But  the  important  thing  is  that  the  solution  of  the 
problem  is  an  act  of  the  pupil's  own  judgment  and  not 
the  application  of  some  meaningless  stereotyped  rule  or 
formula. 


CHAPTER  XV 
PLANNING   THE   LESSON 

NEED  OF  PREPARATION 

BEFORE  conducting  a  class  exercise  of  any  sort,  a  teacher 
should  make  a  careful  preparation  for  the  work  she 
expects  to  do.  She  should  have  definitely .  in  mind  just 
what  she  is  going  to  present  and  why  and  how  she  is  going 
to  present  it.  She  should  understand  the  whole  subject 
so  well  that  she  knows  the  logical  steps  to  follow  in  order 
to  accomplish  her  aim;  that  is,  she  should  know  just 
what  knowledge  the  pupils  have  had  in  order  to  take  up 
the  work  which  she  has  planned  for  the  day. 

THREE  TYPES  OF  LESSONS 

There  are  three  general  types  of  lessons  in  the  teach- 
ing of  arithmetic.  The  work  of  the  day  may  consist  of 
developing  some  new  fact  or  process ;  or  it  may  be  devoted 
to  drilling  upon  previously  developed  facts  or  processes 
in  order  to  give  automatic  control  of  them;  or  it  may 
be  an  application  of  principles  and  processes  already 
learned  and  made  automatic  to  the  solution  of  problems. 
The  nature  of  the  teacher's  part  in  the  class  exercise 
depends  upon  which  of  these  three  phases  is  under  con- 
sideration. These  phases  will  be  considered  under: 

191 


192  THE   TEACHING  OF   ARITHMETIC 

(1)  the  drill  lesson;    (2)  the  inductive  lesson;    and  (3) 
.the  deductive   lesson. 

THE  DRILL  LESSON 

The  pupil  must  have  an  automatic  control  of  all  of 
the  fundamental  facts  and  processes  of  arithmetic.  This 
control  can  be  accomplished  only  through  well-chosen 
motivated  drill.  In  some  schools  there  is  too  great  an 
emphasis  placed  upon  drill.  In  others  there  is  too  little. 
The  teacher  should  consider  carefully  the  three  phases 
of  her  work  in  order  that  they  may  be  well  balanced. 

The  teacher  should  realize,  however,  that  accuracy 
and  a  f air  degree  of  rapidity  in  the  four  fundamental  pro- 
cesses are  of  first  importance,  and  to  secure  this  all  other 
desiderata  must  be  subordinated.  To  secure  and  keep 
this  skill,  constant  practice  and  drill  are  indispensable, 
not  only  in  the  lower  grades,  when  the  fundamental  pro- 
cesses are  first  taught,  but  throughout  the  entire  course. 

NEED  OF  MOTIVATION 

While  drill  exercises  may  seem  mechanical  and  depress- 
ing, there  is  no  substitute  for  them,  and  the  ingenious 
teacher  will  seek  to  make  them  attractive  through  com- 
petitive tests,  by  working  with  a  time  limit  to  make  the 
pupil  conscious  of  his  growing  power,  or  by  other  forms 
of  motivation  suggested  in  the  preceding  chapters. 

It  is  not  sufficient  for  a  child  in  the  lower  grades  to  be 
told  that  his  future  success  depends  upon  the  habits 
developed  by  this  drill.  He  must  have  an  immediate 
motive;  and  the  more  immediate  the  motive  —  the 


PLANNING   THE  LESSON  193 

stronger  the  incentive  —  the  greater  the  attention  that 
will  be  given  to  the  work,  and  the  more  rapid  the  progress 
that  he  will  make. 

DRILL  MUST  INCLUDE  EACH  FACT  OF  A  SERIES 

Often  the  drill  lessons  are  not  systematically  planned. 
They  are  made  up  at  random  and  lack  system.  Easy 
combinations,  for  example,  may  be  given  more  often 
than  more  difficult  ones.  If  the  drills  are  made  up  at 
random  during  the  class  period,  many  of  the  combinations 
may  be  repeated  uselessly  while  others  may  be  omitted 
entirely.  The  class  period  may  have  merely  "  killed  time  " 
and  served  as  little  purpose  as  some  so-called  "  busy  work." 

When  the  object  of  the  drill  is  to  secure  an  automatic 
response  to  a  series  of  facts,  the  teacher  must  see  to  it 
that  every  fact  of  the  series  receives  the  required  atten- 
tion. If  the  drill  is  upon  the  forty-five  primary  facts 
of  addition,  the  teacher  must  make  sure  that  every  fact 
is  included  in  the  drill.  This  does  not  imply  that  each 
fact  should  recur  exactly  the  same  number  of  times. 
Thus,  when  drilling  from  an  addition  chart  like  the  one 
given  in  Chapter  IV,  the  latter  part  of  the  chart  should 
receive  much  more  emphasis  than  the  first  part  of  it. 

Unless  great  care  is  exercised,  the  problems  given  in 
the  lower  grades  for  the  purpose  of  motivating  drill  work 
are  likely  to  make  too  great  use  of  certain  combinations 
to  the  exclusion  of  others.  This  is  necessarily  so,  for  the 
problems  should  represent  real  conditions  as  to  prices, 
sizes,  quantities,  etc.,  and  this  limits  the  numbers  to  be 
used.  For  this  reason,  problems  for  the  purpose  of  fur- 


194  THE    TEACHING  OF   ARITHMETIC 

nishing  drill  work  in  the  fundamental  facts  should  be  used 
but  sparingly,  and  the  teacher  should  exercise  great  care 
in  order  to  get  variety  in  the  combinations  involved. 

As  indicated  above,  the  greater  part  of  the  time  given 
to  a  drill  lesson  should  be  spent  upon  those  facts  or  pro- 
cesses that  present  the  greatest  difficulty.  Frequent 
testing  will  show  the  teacher  which  facts  have  come  under 
the  desired  control  and  which  ones  need  special  drill, 
and  her  drills  should  be  planned  accordingly. 

THE  REPETITION  OF  DRILLS 

When  a  table  is  learned,  it  must  be  drilled  upon  daily 
until  it  can  be  accurately  and  quickly  given.  But  this 
is  not  sufficient.  A  table  may  be  given  ever  so  glibly 
by  a  pupil  after  a  few  days  of  drill,  and  yet  the  facts  that 
are  not  used  for  a  week  or  so  may  be  entirely  forgotten 
when  needed  again.  When  tables  are  first  learned,  the 
periods  between  drills  should  be  short.  Thus,  if  a  pupil 
does  not  have  occasion  to  use  certain  facts  of  addition  for 
several  weeks  after  first  learning  them,  he  may  entirely 
forget  them.  Hence,  as  the  tables  of  a  new  process  are 
learned,  drill  upon  those  already  learned  cannot  be 
neglected.  However,  the  frequency  in  which  they'Tteqes- 
sarily  recur  gradually  changes.  As  the  child  matures, 
periods  between  drills  may  gradually  be  lengthened. 

HOLDING  THE  ATTENTION 

The  results  accomplished  in  any  kind  of  drill  work 
depend  upon  the  degree  of  attention  given  by  the  pupil. 
How  to  secure  a  maximum  of  attention  is  the  vital  problem, 


PLANNING   THE  LESSON  195 

then,  that  confronts  the  teacher.  It  is  difficult  for  chil- 
dren to  fix  then*  attention  for  a  considerable  length  of 
time.  A  teacher  could  hardly  expect  that  a  maximum  of 
attention  could  be  given  a  drill  that  extended  throughout 
the  entire  class  period  without  variation.  When  the 
pupils  show  signs  of  lagging  in  attention,  the  form  of  the 
drill  should  be  changed. 

In  concert  work  a  large  number  of  the  class  are  not 
really  attending  to  the  drills  but  are  saying  what  the 
leaders  say  and  are  thus  getting  no  value  from  the  exer- 
cise. So,  while  some  concert  drill  work  may  occasionally 
be  given,  it  should  not  consume  an  important  part  of  the 
class  period. 

On  the  other  hand,  individual  drill  has  to  be  done  very 
skillfully,  or  the  one  reciting  may  be  the  only  one  paying 
attention  to  the  work  that  is  being  done.  It  frequently 
becomes  necessary,  however,  to  have  an  individual 
pupil  recite  an  entire  series  of  facts ;  but  there  must  be 
some  incentive  to  get  each  pupil  of  the  class  silently  to 
go  through  the  same  recitation.  That  was  the  purpose 
for  which  many  of  the  games  given  in  Chapter  VIII  were 
invented. 

A  method  of  securing  the  maximum  attention  of  a 
class  cannot  be  outlined.  Much  depends  upon  the  per- 
sonality and  the  versatility  of  the  teacher.  She  cannot 
create  interest  and  keep  the  attention  of  the  class  fixed 
upon  the  work  unless  she  too  is  interested  in  it.  A  wide- 
awake, alert,  resourceful  teacher  will  readily  devise  means 
of  securing  attention  as  occasion  for  it  arises;  while  a 
listless  teacher,  or  one  who  lacks  the  power  to  sense  a 


196  THE   TEACHING  OF   ARITHMETIC 

situation,  could  not  hold  the  attention  of  a  class  even 
though  she  knew  all  the  laws  upon  which  attention 
depends. 

While  much  depends  upon  the  resourcefulness  of  the 
teacher,  among  the  various  means  of  securing  attention 
that  may  be  listed  are :  (1)  variety  in  the  forms  of  drill ; 
(2)  working  with  a  time  limit ;  (3)  desire  to  excel  former 
records;  (4)  the  desire  of  a  pupil  to  do  as  well  as  other 
members  of 'the  class ;  (5)  the  desire  of  the  class  as  a  whole 
to  do  as  well  as  or  to  excel  some  former  class ;  or  (6)  the 
desire  of  the  class  to  do  as  well  as  or  excel  a  class  of  the 
same  grade  in  some  other  school. 

THE  INDUCTIVE  LESSON 

Although  much  of  the  time  in  the  primary  grades  is 
spent  in  drill  work,  still  the  teaching  of  arithmetic  fur- 
nishes many  opportunities  to  stimulate  and  train  the  pupil 
to  analyze  a  situation  and  draw  a  conclusion.  The  tradi- 
tional "development  lesson,"  designed  to  stimulate  think- 
ing, is  usually  treated  under  five  formal  steps :  (1)  prep- 
aration ;  (2)  presentation ;  (3)  comparison ;  (4)  generali- 
zation; and  (5)  application.  While  there  is  no  longer  a 
rigid  adherence  to  these  five  steps  followed  in  the  order 
given,  and  without  any  elimination  or  combination,  still 
a  recognition  of  them  will  be  of  help  whenever  the  induc- 
tive development  of  a  fact  or  process  seems  desirable. 
These  five  formal  steps,  however,  must  be  considered 
as  suggestive  rather  than  as  furnishing  a  fixed  mode  of 
procedure. 


PLANNING   THE   LESSON  197 

THE  PEEPARATION 

To  prepare  a  pupil  to  think,  a  problem  whose  solution 
is  necessary  must  first  be  presented.  Thought  is  stimu- 
lated only  when  one  realizes  some  aim  that  is  to  be  satis- 
fied by  the  process.  This  is  formally  called  "develop- 
ing an  aim." 

The  solution  of  this  new  problem,  however,  depends 
upon  some  old  experience;  hence,  another  factor  in  the 
preparation  is  a  review  of  the  related  old  experience. 
These  two  phases  of  the  preparation  :  (1)  the  recognition 
of  a  problem  whose  solution  is  necessary,  and  (2)  a  review 
of  the  old  facts  upon  which  the  solution  depends,  need 
not  follow  each  other  in  this  order.  The  second  phase 
may  even  be  eliminated  when  the  teacher  knows  without 
a  review  that  all  the  old  facts  are  fresh  in  the  pupil's 
mind. 

Thus,  before  a  pupil  takes  up  the  addition  of  two 
fractions  whose  units  are  unlike  as  -g-  +  f ,  he  has  met  and 
solved  the  two  problems  in  the  addition  of  two  fractions, 
the  one  whose  units  are  alike,  and  the  other  in  which 
one  unit  can  be  changed  to  the  other,  as  for  example : 
i  +  f  =  f ,  and  ?  +  i  =  l  +  T  =  -t.  Now,  it  is  not  neces- 
sary to  spend  time  in  reviewing  these  two  problems  if 
they  have  immediately  preceded  this  one,  except  to  show 
wherein  the  new  one  is  different  from  the  old  ones,  and  thus 
get  the  pupil  to  see  the  new  problem  before  him,  which  is 
the  changing  of  fractions  to  like  units. 

To  spend  time  reviewing  well-known  facts  instead  of 
proceeding  at  once  to  the  new  will  cause  a  loss  of  interest. 


198  THE   TEACHING  OF  ARITHMETIC 

THE  PRESENTATION 

After  a  full  realization  of  the  problem  to  be  solved, 
the  next  step  is  to  relate  the  elements  of  the  new  to  facts 
already  known.  No  rule  can  be  laid  down  that  will  fully 
cover  the  teacher's  part  in  this.  Sometimes,  through 
a  series  of  questions,  she  gets  the  pupil  to  relate  the  new 
with  the  old.  At  other  times  she  finds  it  more  economical 
to  illustrate  or  demonstrate  the  relations  and  ask  fewer 
questions.  Thus,  the  presentation  of  the  area  of  a  circle 
requires  many  more  directions  and  demonstrations  than 
mere  questions.  After  the  pupil  has  realized  that  the 
circle  cannot  be  divided  into  a  number  of  square  inches, 
as  in  the  case  of  the  rectangle,  and,  as  it  stands,  it  cannot 
be  divided  up  into  triangles,  parallelograms,  or  trapezoids, 
as  all  plane  rectilinear  figures  could,  he  has  grasped  the 
problem.  At  least  he  sees  that  here  is  an  area  to  be  meas- 
ured that  differs  in  form  from  the  others  and  seems  to 
demand  a  new  "rule."  Now  in  the  presentation  the 
teacher  can  show  that  by  dividing  the  circle  into  a  number 
of  equal  sectors  and  properly  fitting  them  together,  the 
circle  may  be  converted  into  what  is  practically  a  rectangle 
or  parallelogram  with  known  dimensions.  Now  through 
a  series  of  questions  she  gets  from  the  pupil  that  the  area 
must  equal  that  of  a  rectangle  whose  base  is  half  the 
circumference  and  whose  width  is  the  radius,  which  is 
really  the  next  two  of  the  formal  steps  —  the  comparison 
and  the  generalization. 

Avoid  questions,  however,  that  are  so  suggestive  that 
they  fail  to  stimulate  thought. 


PLANNING   THE   LESSON  199 

THE  COMPARISON 

With  the  problem  before  us  and  the  necessary  data 
furnished,  the  pupil  is  led  to  compare  the  new  with  what 
is  already  known.  Thus  in  the  illustration  given  above, 
in  finding  the  area  of  a  circle,  the  pupil  recognizes  the 
problem  and  is  furnished  with  a  circle  cut  into  sixteen  or 
more  equal  sectors  which  are  rearranged  to  somewhat 
resemble  a  rectangle.  The  pupil  observes  several  such 
circles  in  their  new  arrangement  and  compares  the  forms 
they  now  assume  and  draws  the  conclusion  that  they  are 
equivalent  to  rectangles  with  known  dimensions. 

THE  GENERALIZATION 

After  making  comparisons  and  forming  conclusions 
until  satisfied  of  certain  fixed  facts  that  hold  good  in  all 
cases,  the  pupil  summarizes  these  facts  into  a  definition 
or  principle,  or  into  a  working  rule  for  later  use.  Thus, 
in  the  example  given,  the  pupil  generalizes  his  conclusions 
and  states  that  the  area  of  a  circle  is  equal  to  that  of  a 
rectangle  whose  base  is  half  the  circumference  and  whose 
width  is  the  radius.  Or,  stated  in  the  form  of  a  rule,  to 
find  the  number  of  units  hi  the  area  of  a  circle,  multiply 
the  number  of  units  in  half  the  circumference  by  the 
number  in  the  radius. 

Through  the  aid  of  the  teacher,  the  rule  is  then  fitted 
into  the  conventional  form  to  be  memorized,  as : 


200  THE   TEACHING  OF   ARITHMETIC 

Besides  aiding  directly  in  forming  habits  of  right 
thinking,  the  discovery  and  generalization  of  facts  and 
processes  by  the  pupil  for  himself  give  him  a  better 
grasp  of  the  subject  so  that  it  is  used  more  intelligently 
and  remembered  much  more  easily.  The  mere  ability 
to  repeat  parrot-like  the  rules  or  definitions  of  arithmetic 
with  no  concrete  background  as  a  basis  for  them  is  prac- 
tically valueless. 

THE  APPLICATION 

When  conclusions  have  been  reached  and  generaliza- 
tions made,  opportunity  for  them  to  be  usecl  in  some 
practical  problem  must  be  furnished.  In  so  far  as  possible, 
the  applications  found  in  the  textbook  must  be  supple- 
mented by  problems  that  some  one  in  the  life  about  them 
needs  to  solve. 

SOME  CAUTIONS  IN  USING  THE  INDUCTIVE  METHOD 

A  teacher  in  planning  the  development  of  a  rule  or 
principle  must  not  feel  that  in  order  to  do  it  properly 
the  five  formal  steps  must  all  be  followed  as  five  distinct, 
exclusive  steps.  Often  there  is  no  sharp  differentiation 
between  them.  In  seeking  to  bring  out  the  five  steps, 
she  may  make  the  work  formal  and  uninteresting.  Long, 
tedious  presentations  should  be  avoided.  If  a  truth  has 
been  grasped  and  can  be  applied,  that  is  the  supreme 
test. 

Do  not  give  too  much  help.  Some  teachers  interpret 
the  inductive  method  to  mean  that,  through  a  series  of 
carefully  planned  questions,  the  pupil  is  brought  to  state 


PLANNING   THE  LESSON  201 

the  generalization.  Often  these  questions  are  so  suggestive 
as  to  require  no  thought  upon  the  part  of  the  pupil.  After 
the  problem  is  fully  presented  to  the  pupil,  the  more  he 
does  for  himself,  the  better.  It  is  the  teacher's  part  to 
stimulate  the  pupil  to  seek  the  solution,  but  to  give  as 
few  hints  and  to  ask  as  few  leading  questions  as  possible. 

THE  FORMAL  STEPS  IN  TEACHING  THE  AREA  OP  A 
RECTANGLE 

This  is  taken  up  here  to  illustrate  the  five  formal  steps 
of  an  inductive  lesson  that  have  just  been  discussed.  By 
referring  to  the  method  of  developing  the  various  topics 
given  in  former  chapters  the  teacher  should  be  able  to 
thus  arrange  the  steps  in  the  teaching  of  any  topic. 

Preparation.  —  Recall  the  methods  of  measuring  other 
magnitudes,  as  lines,  the  contents  of  vessels,  etc.  See 
that  the  pupil  understands  that  to  measure  any  magnitude 
is  to  find  how  many  of  some  standard  unit  it  will  contain. 
Show  some  standard  units  used  in  measuring  surfaces, 
as  a  square  inch,  a  square  foot,  etc.  Show  some  rectangle 
whose  dimensions  are  integers  and  measure  it  by  placing 
enough  square  units  on  it  to  cover  it.  Raise  the  question 
of  how  the  number  needed  to  cover  it  could  have  been 
calculated  so  as  to  save  the  necessity  of  covering  the  area 
with  square  units.  The  problem  is  thus  fully  presented ; 
that  is,  the  teacher  has  developed  "the  pupil's  aim." 

Presentation.  —  Furnish  pupils  with  rectangles  of  card- 
board and  with  but  one  square  unit  each  with  which  to 
measure  them.  Let  them  find  the  area  in  any  way  they 
can.  Encourage  them  to  thus  lay  off  the  rectangles  into 


202  THE   TEACHING  OF   ARITHMETIC 

square  units.  Ask  them  to  compare  the  number  of  square 
units  in  one  row  along  the  length  with  the  number  of 
linear  units  in  the  length.  Then  ask  them  to  compare  the 
number  of  such  rows  with  the  number  of  linear  units  in 
the  width. 

Comparison.  —  Have  the  pupils  make  several  compari- 
sons as  above  and  draw  conclusions  until  they  can  tell 
without  drawing  that  a  rectangle  3  in.  by  4  in.  is 
composed  of  3  rows  of  4  sq.  in.  each,  that  one  5  in.  by 
6  in.  is  composed  of  5  rows  of  6  sq.  in.  each,  etc.  From 
this  have  them  see  that  in  a  rectangle  3  in.  by  4  in. 
there  are  3X4  sq.  in.  In  one  5  in.  by  6  in.  there  are 
5X6  sq.  in.,  etc. 

Generalization.  —  From  the  conclusions  formed  above, 
have  the  pupils  formulate  the  fact  that  the  number  of 
square  units  in  the  area  of  a  rectangle  is  the  product  of 
the  number  of  linear  units  in  the  two  dimensions. 

Application.  —  Test  the  pupils'  understanding  of  the 
principle  just  stated  by  applying  it  to  problems  of  the 
textbook,  then  to  those  found  out  of  the  text.  In  measur- 
ing rectangles,  select  those  whose  measurement  is  neces- 
sary. Thus,  how  much  did  the  walk  in  front  of  the  school 
cost?  Get  the  local  price  per  unit  and  thus  answer  the 
question.  There  is  an  abundance  of  real  problems  on 
every  hand  that  require  the  measurement  of  rectangles. 

THE  LESSON  PLAN 

In  writing  out  a  lesson  plan  embracing  the  five  steps 
as  given  above,  it  is  customary  to  ask  questions  to  show 
the  "teacher's  part"  and  then  answer  them  to  show  the 


PLANNING   THE  LESSON  203 

"pupil's  part."  While  a  teacher  should  have  clearly  in 
mind  just  the  questions  that  she  intends  to  ask,  she  must 
realize  that  the  pupils  will  not  always  answer  as  she  has 
expected  them  to  do  and  hence  that  her  questions  will 
have  to  be  changed.  Aside  from  the  general  questions, 
then,  she  must  fully  realize  what  each  step  consists  of 
and  how  she  expects  to  attain  the  ends  sought  and  then 
secure  the  greatest  possible  self-activity  from  the  pupil. 
Questions  are  often  so  leading  that  the  pupil  has  to  do 
but  little,  if  any,  thinking.  This  should  be  carefully 
avoided.  The  supreme  test  of  teaching  is  the  ability 
of  the  teacher  to  arouse  the  self-activity  of  the  pupil. 

THE  DEDUCTIVE  LESSON  f" 

The  complete  thought  process  involves  both  the  induc- 
tive and  deductive  types  of  thinking.  A  lesson  is  rarely 
ever  purely  inductive  or  purely  deductive.  All  laws  are 
first  arrived  at  through  the  process  of  induction;  but, 
in  verifying  them,  the  thought  is  deductive.  The  present 
plan  of  developing  the  principles  of  arithmetic  is  more 
inductive  than  deductive.  The  applications  of  these 
principles  are  more  deductive.  However,  in  teaching  a 
process  or  rule,  the  plan  sometimes  becomes  more  deduc- 
tive than  inductive.  Thus,  in  teaching  the  rule  for 
finding  the  area  of  a  rectangle  given  under  the  inductive 
plan,  had  the  teacher  equipped  the  class  with  rectangles 
ruled  in  squares  or  with  squared  paper  and  then  asked 
them  to  verify  the  fact  that  the  number  of  square  units 
was  equal  to  the  product  of  the  number  of  linear  units 


204  THE   TEACHING  OF  ARITHMETIC 

in  the  two  dimensions,  the  process  would  have  been  de- 
ductive. Whether  the  teaching  process  is  to  be  inductive 
or  deductive,  the  first  step  is  the  preparation;  that  is, 
needed  knowledge  is  reviewed  and  the  problem  fully 
presented.  A  generalization  is  then  assumed,  and,  by 
appealing  to  known  facts,  it  is  verified  through  a  course 
of  reasoning. 

In  the  solution  of  a  problem,  the  type  of  reasoning 
is  largely  deductive.  Thus,  there  is  first  a  clear  recogni- 
tion of  the  problem.  Then  follows  an  analysis  of  condi- 
tions in  seeking  the  principles  to  be  applied.  In  general, 
it  is  more  difficult  to  select  and  classify  the  formal  steps, 
particularly  in  the  solution  of  a  problem,  than  it  was  in 
the  inductive  type.  Thus,  if  one  is  to  find  a  selling  price 
of  an  article  costing  $2.10  that  will  give  a  profit  of  30% 
of  itself,  the  problem  is  first  studied,  and  it  is  clearly 
seen  that  this  problem  is  one  in  which  the  basis  upon 
which  the  per  cent  is  reckoned  is  not  given,  but  is  wanted. 
Hence,  it  cannot  be  the  first  type  of  problem.  Then  one 
reasons  that,  since  30%  of  the  selling  price  must  be 
profit,  the  remainder  of  it,  or  70%  of  it,  must  represent 
the  cost.  Hence,  the  relation  70%  of  the  selling  price 
equals  $2.10  is  seen.  Now,  if  this  type  of  relation  is  a 
seemingly  new  one,  the  reasoning  again  is  somewhat  as 
follows:  This  means  .70 X  an  unknown  selling  price  = 
$2.10.  But  this  means  that  the  product  of  two  factors 
and  one  of  the  factors  is  known  and  the  problem  is  to 
find  the  other.  But  from  the  meaning  of  division  the 
solution  must  be  $2. 10 -f-. 70.  Hence,  the  selling  price 
must  be  $3.  To  verify  this,  the  gam  must  be  30%  of 


PLANNING   THE  LESSON  205 

$3  or  90  cents.  And  this  leaves  $2.10  for  the  cost  as  it 
should.  Hence,  the  solution  is  correct.  Thus,  it  is  seen 
that  there  is  a  series  of  problems  arising,  each  followed 
by  an  analysis  and  a  conclusion  of  which  principles  to 
apply,  and  finally  a  verification  of  the  conclusion. 


CHAPTER  XVI 
THE  COURSE   OF  STUDY  IN  ARITHMETIC 

ONE  of  the  chief  factors  in  efficient  school  work  is  the 
course  of  study.  And,  while  it  must  not  be  inflexible, 
yet  it  must  be  based  upon  certain  fixed  fundamental 
principles.  Each  year's  work  must  be  based  as  nearly 
as  possible  upon  the  needs,  interests,  experiences,  and 
personality  of  the  learner,  and  made  to  function  with  his 
life  both  in  and  out  of  school.  That  is,  the  material  used 
must  be  based  upon  some  social  issue  of  present  or  near 
future  value  and  not  given  for  the  mere  purpose  of  training. 

There  are  three  natural  divisions  of  the  course  in  arith- 
metic, each-  with  its  own  aims,  methods,  and  material 
all  rather  distinctly  marked.  These  three  divisions  fall 
naturally  under  the  so-called  Primary,  Intermediate, 
and  Grammar  School  grades.  Textbook  makers  have 
followed  this  division  in  their  texts,  and  most  modern 
textbooks  thus  become  a  sort  of  course  of  study.  But 
the  matter  of  any  textbook  must  be  adapted  to  the  ends 
to  be  attained  by  omitting  and  supplementing  to  meet  the 
needs  of  the  child,  for  it  must  not  be  forgotten  that  it 
is  the  child  and  not  the  subject  that  is  being  developed. 
It  is  his  temperament,  his  interests,  his  needs,  and  his 
abilities  that  must  be  studied,  and  the  instruction  must 
be  shaped  to  fit  them. 

206 


THE   COURSE  OF  STUDY   IN  ARITHMETIC    207 

THE   PRIMARY   GRADES 

The  primary  grades  include  the  work  of  the  first  four 
years.  The  aims  of  these  grades  are  clearly  defined  and 
definite.  They  are :  (1)  to  teach  the  reading  and  writing 
of  numbers;  (2)  to  give  an  automatic  control  of  the 
primary  number  facts  (the  so-called  tables) ;  and  (3)  to 
develop  some  skill  in  the  written  processes  with  whole 
numbers.  While,  of  course,  all  these  will  find  applica- 
tions within  the  child's  interests  and  needs,  the  problems 
of  the  primary  grades  are  of  minor  importance  except  in 
so  far  as  they  make  clear  the  meaning  of  the  processes  and 
motivate  the  drill  work.  Neither  is  the  rationalization 
of  the  fundamental  processes  with  whole  numbers  of 
importance.  It  is  proper  habits  of  procedure  that  are  of 
chief  importance.  Thus  the  chief  problem  of  the  teacher 
of  these  grades  is  how  to  bring  about  a  control  of  the 
primary  facts  and  dependable  habits  of  procedure  in  the 
written  processes  with  the  greatest  economy  of  time  and 
effort.  The  problem,  then,  is  a  study  of  methods  of  drill 
and  of  the  proper  gradation  of  the  written  work  —  a  study 
of  the  means  of  vitalizing  the  work  so  as  to  secure  a  maxi- 
mum of  attention.  These  have  all  been  discussed  in 
former  chapters  of  this  book. 

THE  FIRST  GRADE 

Following  the  principle  that  the  work  of  each  year 
shall  as  nearly  as  possible  be  based  upon  the  child's  needs 
and  interests,  there  is  a  rapidly  growing  tendency  to  defer 
drill  in  formal  number  work  until  the  second  school  year. 


208  THE   TEACHING  OF  ARITHMETIC 

However,  there  are  needs  of  counting  and  of  expression 
during  the  first  year  that  should  be  met.  In  most  locali- 
ties the  child's  needs  during  the  first  year  are  limited  to 

(1)  reading  numbers  to  100,  or  possibly  to  reading  num- 
bers of  three  figures,  in  order  to  find  a  page  in  his  book  or 
the  number  on  a  house,  if  he  lives  in  a  city ;   (2)  reading 
the  Roman  numerals  to  XII  on  a  dock  face,  in  order  to 
tell  time ;   (3)  counting  by  ones  and  tens  to  100,  in  order  to 
count  the  words  or  lines  in  a  lesson  or  the  objects  with 
which  he  works  in  other  school  activities;    (4)  possibly 
the  recognition   of   one   half,   one  third,  and  one  fourth 
of  an  object  that  has  been  divided ;  and  also  (5)  famil- 
iarity with  the  common  units  of  measure  used  by  him  in 
or  out  of  school.    All  this  may  be  done  in  connection  with 
other  school  work,  without  setting  aside  a  period  in  the 
program  especially  for  number  work. 

THE  SECOND  GRADE 

The  work  of  this  grade  should  be  entirely  oral.  The 
most  important  aims  of  the  year's  work  are :  (1)  to  make 
automatic  the  forty-five  primary  number  facts  of  addi- 
tion and  the  corresponding  eighty-one  subtraction  facts ; 

(2)  to  develop  some  skill  in  calling  the  sum  of  any  two- 
figured  number  added  to  a  one-figured  number;   and  (3) 
to  develop  some  ability  in  adding  three  or  four  one-figured 
numbers.     Compared  with  these  three  aims,  the  other 
arithmetic  work  of  the  year  is  of  minor  importance. 
At  the  end  of  the  second  year,  then,  there  should  have 
been  accomplished  the  following: 

1.  Reading  and  writing  numbers  to  1000. 


THE  COURSE  OF  STUDY  IN  ARITHMETIC    209 

2.  Reading  Roman  numerals  to  XII. 

3.  Telling  time  to  hours  and  half  hours. 

4.  Reading  and  writing  dollars  and  cents. 

5.  Interpreting  the  signs  +,  — ,  =,$,£. 

6.  The  forty-five  primary  facts  of  addition  and  the 
corresponding  eighty-one  subtraction  facts. 

7.  Some  skill  in  adding  at  sight  any  one-figured  num- 
ber to  any  two-figured  number. 

8.  Some  skill  in  adding  single  columns  of  three  or 
four  one-figured  numbers. 

9.  Recognition  of  halves,  thirds,  and  fourths  of  single 
objects. 

10.  Those  measuring  units  used  in  any  of  the  child's 
activities  either  hi  or  out  of  school. 

SUGGESTIONS.  —  1.  Careful  instruction  and  much  practice 
in  making  the  figures  and  signs  should  be  given,  so  that  in  later 
written  work  the  making  of  these  will  not  retard  thinking. 
While  all  the  work  of  this  grade  is  oral,  the  pupil  should  spend 
much  time  in  copying  figures  and  his  "tables"  just  as  he  copies 
letters  and  words. 

2.  Addition  and  subtraction  should  be  treated  as  correlative 
processes.     Thus,  when  a  child  learns  that  3  and  5  are  8,  he 
learns  that  either  part  taken  from  8  leaves  the  other. 

3.  Drill  with  charts  and  flash  cards  until  all  the  combinations 
can  be  given  instantly  at  sight. 

4.  Observe  from  Chapter  IV  that  the  forty-five  primary  num- 
ber facts  of  addition  are  more  economically  treated  by  dividing 
them  into  two  groups:     (1)     those  twenty-five  combinations 
whose  sums  do  not  exceed  10;    and  (2)  those  twenty  whose 
sums  do  exceed  10.     The  sums  of  the  first  group  are  found 
through  counting  in  order  to  fix  the  meaning  of  addition.     The 
sums  of  the  second  group  are  not  found  through  counting.     A 
child  is  not  able  to  image  eight  or  nine  things,  much  less  their 
sum.     No  aid,  then,  to^memorizing  the  second  group  is  secured 


210  THE    TEACHING  OF   ARITHMETIC 

through  finding  the  sums  through  counting,  and  the  finding  of 
the  first  group  is  sufficient  to  fix  the  meaning  of  addition.  (See 
Ch.  IV.) 

5.  The  drills  should  be  written  in  column  form  as  the  child 
is  to  see  them  later  in  written  work.     Sight  work  should  precede 
dictation  work,  but  both  forms  of  drill  should  be  used. 

6.  Make  much  use  of  games  to  vitalize  the  work  and  secure 
the  maximum  of  attention. 

THE  THIRD  GRADE 

The  pupil  should  have  a  textbook  in  this  grade  and 
written  work  begins  here.  The  most  important  aims  of 
this  year  are :  (1)  to  develop  some  skill  in  written  addi- 
tion and  subtraction ;  (2)  to  make  automatic  the  primary 
number  facts  of  multiplication  and  the  corresponding 
division  facts;  and  (3)  to  teach  written  multiplication 
and  division  by  one-figured  multipliers  and  divisors, 
the  dividend  being  limited  to  an  exact  multiple  of  the 
divisor.  Compared  with  these,  the  other  aims  of  the 
year  are  of  minor  importance.  In  all,  the  work  of  the 
year  should  include : 

1.  Reading  and  writing  numbers  to  10,000. 

2.  Reading  and  writing  Roman  numerals  to  XII. 

3.  Reading  and  writing  unit  fractions  £,  £,  i,  to  £. 

4.  Review,  of  the  primary  facts  of  addition  and  subtrac- 
tion for  accuracy  and  rapidity. 

5.  More   practice   in   adding  one-figured   numbers  to 
two-figured  numbers. 

6.  More  practice  in  adding  at   sight   single   columns 
of  three  or  four  one-figured  numbers  and  extended  to  in- 
clude five  or  six  one-figured  numbers. 

7.  Written  addition  limited  to  five  or  six  addends-. 


THE   COURSE   OF  STUDY  IN   ARITHMETIC    211 

8.  Written  subtraction. 

9.  The  primary  facts  of  multiplication  and  the  corre- 
sponding division  facts. 

10.  Written  multiplication  and  division  when  the  mul- 
tipliers and  divisors  are  one-figured  numbers. 

SUGGESTIONS.  —  1.  Facility  in  reading  and  writing  numbers 
of  three  or  four  orders  should  come  largely  incidentally  from  the 
ordinary  use  of  numbers,  not  from  teaching  and  drilling  upon 
"notation  and  numeration"  as  separate  topics. 

2.  The  fractions  arise  in  the  partition  phase  of  division  and 
should  be  confined  to  unit  fractions. 

3.  Drill  upon  the  primary  facts  of  the  second  grade  should 
be  continued  until  all  are  recognized  instantly. 

4.  Children  should  not  be  given  much  column  addition  until 
they  can  add  two-figured  and  one-figured  numbers  with  con- 
siderable skill. 

5.  Pupils  need  not  understand  the  "why"  of  carrying  or 
borrowing  in  this  grade.     The  "how"  is  the  important  thing. 

6.  Pupils  should  make  their  multiplication  tables  through 
adding  equal  addends  in  order  to  see  what  multiplication  means. 

7.  Include  any  unit  of  measure  needed  by  the  pupil  to  inter- 
pret his  other  work,  using  the  unit  itself  as  it  is  being  taught. 

8.  All  applications  should  be  kept  within  the  range  of  the 
pupil's   experiences   and   interests.     Dramatized   activities   as 
playing  store,  etc.,  are  much  better  for  this  grade  than  problems 
from  the  world  of  adults. 

THE  FOUKTH  GRADE 

The  greater  part  of  the  time  of  this  year  is  devoted  to 
securing  greater  skill  in  the  work  begun  in  the  third  year. 
The  new  work  of  the  year  consists  chiefly  in  long  multi- 
plication and  long  division,  using  multipliers  and  divisors 
of  two  or  three  figures.  At  the  end  of  the  fourth  grade 
the  work  finished  should  include : 


212  THE   TEACHING  OF  ARITHMETIC 

1.  Reading  and  writing  numbers  to  100,000,000. 

2.  Reading  and  writing  any  of  the  Roman  numerals. 

3.  Skill  in  written  addition  and  subtraction  and  power 
to  check  results  so  as  to  turn  in  work  100%  accurate. 

4.  Complete  mastery  of  the  multiplication  and  division 
tables. 

5.  Skill  in  written  multiplication  and  division  where 
multipliers  and  divisors  do  not  exceed  three-figured  num- 
bers, and  power  to  check  work  in  order  to  turn  in  work 
that  is  100%  accurate. 

6.  Power  to  apply  the  knowledge  thus  far  acquired 
to  problems  that  come  within  the  pupils'   needs  and 
experience. 

SUGGESTIONS.  —  1.  Do  not  perplex  pupils  with  drill  in  "no- 
tation." The  reading  of  numbers  is  of  more  importance.  Facil- 
ity in  reading  and  writing  numbers  com.es  incidentally  through 
the  ordinary  use  of  numbers  rather  than  through  special  drill. 

2.  The  reading  and  writing  of  the  Roman  numerals  is  of  minor 
importance  and  should  not  consume  time  needed  for  more  impor- 
tant parts  of  the  year's  work. 

3.  All  drills  of  the  preceding  grades  should  be  continued  to 
give  greater  skill. 

4.  Confine  the  work  of  denominate  numbers  to  the  needs  of 
the  pupil.     When  a  new  unit  is  introduced  it  should  be  shown 
the  class.     If  square  and  cubic  units  are  taken  up,  see  that 
pupils  get  a  clear  conception  of  them  and  their  use. 

<  THE  INTERMEDIATE   GRADES 

This  includes  the  work  of  the  fifth  and  sixth  grades. 
The  teaching  in  these  grades  takes  on  new  phases.  Ra- 
tionalization plays  a  much  more  important  part.  The 
processes  with  fractions  and  decimals  recur  less  frequently 


THE   COURSE  OF  STUDY  IN   ARITHMETIC     213 

than  those  with  whole  numbers,  hence  rationalization  is 
needed  hi  order  to  help  the  memory  retain  the  "how" 
of  the  processes.  Not  only  on  that  account,  but  because 
the  rationalization  of  the  subject  of  fractions  and  deci- 
mals develops  greater  power  to  use  the  subjects  properly 
when  applied  to  problems  is  rationalization  urged. 

The  applications  of  arithmetic  to  problems  becomes 
much  more  important  in  these  grades  than  in  the  pri- 
mary grades.  The  problems  are  not  given  now  merely  to 
furnish  occasions  for  calculation,  but  they  are  connected 
more  with  some  social  issue  of  interest  to  the  pupil  and 
thus  serve  to  develop  the  habit  of  looking  upon  the  quanti- 
tative side  of  life.  He  begins,  too,  in  these  grades  to  get 
some  insight  into  the  use  of  arithmetic  hi  commercial 
and  industrial  phases  of  the  world's  work. 

THE  FIFTH  GRADE 

Drill  in  the  work  of  the  preceding  grades  should  continue 
in  order  to  secure  greater  skill.  Greater  use  of  the  work 
of  whole  numbers  in  problems  should  be  made.  But 
the  new  work  of  this  year  is  chiefly  the  development  of 
the  fundamental  processes  with  fractions  and  mixed 
numbers  and  application  of  these  processes  to  problems 
that  need  solution.  The  complete  work  of  the  year 
should  include  the  following : 

1.  Review  and  extend  the  work  in  whole  numbers. 

2.  Develop  multiplication  where  the  multiplier  con- 
tains one  or  more  zeros. 

3.  Take  up  division  in  which  zeros  occur  in  the  quotient. 

4.  Teach  the  meaning  of  abstract  and  concrete  numbers. 


214  THE   TEACHING  OF   ARITHMETIC 

5.  Teach  all  four  fundamental  processes  with  fractions 
and  mixed  numbers. 

6.  Teach  the  meaning  of  ratio  and  use  ratio  within 
the  limits  of  familiar  numbers. 

7.  Teach   the   measurement    of    areas    of   rectangles, 
parallelograms,  and  triangles. 

8.  Teach  the  surfaces  and  volumes  of  right  prisms. 

SUGGESTIONS.  —  1.  In  the  applications  do  not  allow  a  pupil 
to  speak  of  the  concrete  factor  as  the  multiplier. 

2.  Develop  the  notion  of  a  fraction  objectively. 

3.  Be  careful  in  developing  the  notation  of  a  fraction  that 
the  pupil  sees  the  use  of  each  term.     The  development  of  the 
processes  depends  upon  the  notation. 

4.  Show  that  as  long  as  the  meaning  of  a  process  does  not 
change,  the  method  does  not  change. 

5.  Relate  all  processes  in  fractions  with  their  corresponding 
processes  with  integers. 

6.  Have  pupils  see  clearly  that  the  so-called  multiplication 
by  a  fraction  is  a  new  meaning  of  multiplication,     fxf  does 
not  mean  "f  times  f "  but  "f  of  £." 

7.  Seek  local  applications  for  all  processes  as  fast  as  learned. 

THE  SIXTH  GKADE 

Fractions,  whole  numbers,  and  mensuration  are  taken 
up  in  this  grade  for  review  and  extended  in  their  applica- 
tions. The  new  work  of  this  grade  consists  of  decimals 
and  percentage.  These,  then,  demand  the  greatest  care 
in  presentation.  The  outline  of  the  year's  work  is  as 
follows : 

1 .  Teach  the  decimal  place-value  feature  of  our  notation. 

2.  Continue    drill    in    whole    numbers,    using    larger 
numbers. 


THE   COURSE  OF   STUDY   IN   ARITHMETIC    215 

3.  Apply  addition  and  subtraction  to  keeping  accounts. 

4.  Teach  short  methods  of  multiplication  and  division, 
using  aliquot  parts  and  special  fractions. 

5.  Review  and  extend  the  meaning  and  use  of  com- 
mon fractions. 

6.  Develop  cancellation. 

7.  Develop  the  meaning  of  ratio  and  how  to  express 
a  ratio  as  a  fraction. 

8.  Teach  the  fundamental  processes  with  decimals. 

9.  Express  any  ratio  in  decimal  form. 

10.  Develop  per  cent  as  a  new  name  and  notation  for  a 
special  fraction. 

11.  Study  the  application  of  percentage,  confining  the 
work   to   the  two  direct    applications   of   the   subject : 
(1)  finding   a  per  cent  of  a   number;  and  (2)  finding 
what  per  cent  one  number  is  of  another. 

12.  Review  the  mensuration  given  in  the  fifth  grade  and 
extend  its  applications. 

SUGGESTIONS.  —  1.  In  this  final  study  of  fractions  seek  to 
develop  the  full  meaning  and  use  of  a  fraction :  (1)  one  or  more 
of  the  several  parts  of  some  whole;  (2)  an  expressed  division; 
and  (3)  the  expression  of  the  ratio  of  one  magnitude  to  another. 

2.  Develop  the  decimal  fraction  as  an  extension  of  our  deci- 
mal place-value  notation  to  the  right  of  ones'  place,  not  as  a 
special  notation  of  a  common  fraction. 

3.  It  is  very  important  that  a  pupil  be  able  to  express  the 
ratio,  as  a  common  fraction,  between  any  two  magnitudes,  then 
change  this  ratio  to  a  decimal,  for  this  is  the  foundation  of  one 
of  the  most  important  problems  of  percentage. 

4.  Present  percentage  as  a  problem  of  decimals,  the  notation 
being  the  only  new  feature. 

5.  But  two  problems  of  percentage  should  be  taught  this 
year  —  to  find  a  per  cent  of  a  number,  and  to  find  what  per  cent 


216  THE   TEACHING  OF  ARITHMETIC 

one  number  is  of  another.  Make  clear  that  both  of  these  prob- 
lems were  encountered  in  decimals,  —  multiplying  by  a  decimal, 
and  expressing  a  ratio  as  a  decimal. 

6.  Nothing  is  gained  in  clearness  of  thought  by  using  the 
terms  "  base,"  "rate,"  and  "  percentage,"  terms  still  used  in  some 
textbooks.     Hence  it  is  better  not  to  use  them  except  when 
"rate"  is  used  in  such  expressions  as  "rate  of  gain,"  "rate  of 
interest,"  "rate  of  discount,"  etc. 

7.  While  the  applications  now  begin  to  take  on  more  and 
more  of  the  adult  point  of  view,  keep  them  well  within  the  pupil's 
experiences.     Encourage   pupils    to    bring   in    advertisements, 
sample  business  papers,  bills,  accounts,  notes,  etc.     Make  the 
commercial  applications  more  real  and  concrete  by  dramatiz- 
ing the  adult  activities  that  are  studied.     Children  like  to  play 
"  Going  into  business." 

8.  In  local  applications  of  mensuration  have  the  pupils  make 
their  own  measurements  in  getting  the  needed  data. 

THE   GRAMMAR   SCHOOL  GRADES 

This  includes  the  work  of  the  seventh  and  eighth 
grades.  Educational  opinion  seems  less  settled  upon 
what  should  be  included  in  these  two  grades  than  in  the 
earlier  grades.  In  most  schools  the  work  still  consists 
almost  entirely  of  arithmetic.  In  schools  organized 
under  the  Junior  High  School  plan  some  algebra  and 
observational  geometry  are  being  introduced  in  these 
grades  and  arithmetic  continued  into  the  ninth  grade. 

It  seems  clear,  however,  that  in  so  far  as  the  maturity 
of  the  pupil  will  permit,  the  work  of  these  two  grades 
should  furnish  the  mathematics  needed  by  the  average 
intelligent  citizen  outside  of  a  specialized  vocational 
need.  The  needs  of  such  a  person  are  (1)  power  to  see 
and  express,  and  to  interpret  the  expressions  of  the 
quantitative  relations  that  come  within  one's  needs  and 


THE   COURSE  OF  STUDY  IN  ARITHMETIC    217 

interests ;  (2)  the  habit  of  looking  upon  the  quantitative 
side  of  life  and  seeing  these  relations,  particularly  those 
vital  to  one's  welfare;  and  (3)  a  knowledge  of  commer- 
cial and  industrial  practices  through  which  one  may 
interpret  references  met  in  general  reading  and  in  one's 
business  and  social  intercourse  with  those  with  whom  he 
comes  in  contact. 

With  these  aims,  then,  as  the  basis  of  the  course,  the 
subject  is  organized  around  some  social  topic  instead  of 
around  some  arithmetical  topic,  as  borrowing  and  loan- 
ing money,  buying  stocks,  bond  investments,  taxes,  insur- 
ance, etc.  Or,  these  are  sometimes  included  as  parts  of 
larger  units,  as  problems  of  thrift,  problems  of  investment, 
problems  of  protection,  problems  of  public  expense,  prob- 
lems of  transportation,  problems  of  industrial  life,  etc. 
The  algebra  taught  in  these  grades  is  usually  limited  to 
the  interpretation  of  the  formula  and  to  the  use  of  the 
simple  equation.  The  mensuration  of  the  past  is  now 
sometimes  supplemented  by  a  little  constructive  and 
observational  geometry  and  listed  under  the  general 
topic  of  "geometry."  Regardless  of  the  grouping,  the 
following  topics  are  usually  included  in  these  two  grades : 

THE  SEVENTH  GRADE 

1.  A  review  of  whole  numbers,  fractions,  and  decimals, 
including  short  methods  and  emphasis  upon  checking 
results  and  estimating  results. 

2.  The  two  problems  of  percentage  that  were  taken 
up  in  the  sixth  grade  as  applied  to  profit  and  loss;  dis- 
count ;   commission ;   simple  interest ;   bank  discount. 


218  THE   TEACHING  OF   ARITHMETIC 

3.  The  measurement  of  the  areas  of  rectangles,  tri- 
angles, parallelograms,  trapezoids,  and  circles;    and  the 
measurement  of  the  surfaces  and  volumes  of  prisms  and 
cylinders  as  applied  to  practical  problems. 

4.  Drawing  to  a  scale  and  sketching  and  interpreting 
plans  and  diagrams. 

SUGGESTIONS.  —  1.  Attention  this  year  to  the  applications 
of  arithmetic  is  the  important  thing.  These  applications  in- 
clude denominate  numbers,  mensuration,  and  percentage. 

2.  Confine  the  work  of  percentage  to  the  two  practical  prob- 
lems of  the  sixth-grade  outline.     Fix  in  memory  all  the  impor- 
tant fractional  equivalents  by  constant  use  and  drill. 

3.  Seek  to  make  all  work  applied  to  commercial  transactions 
as  realistic  as  possible.     Play  "Going  into  business"  if  it  adds 
interest  and  makes  the  work  more  realistic.     All  commercial 
schools  do  this  with  students  of  much  greater  maturity. 

4.  Encourage  pupils  to  bring  in  real  problems,  commercial 
papers,  and  any  material  that  will  vitalize  the  work. 

5.  Distinguish  between  real  problems  met  in  real  life,  and 
problems  "for  analysis"  about  real  things.     Remember  that  a 
"real"  problem  is  not  necessarily  concrete  to  the  pupil. 

6.  In  the  application  of  mensuration,  encourage  pupils  to 
bring  in  real  problems  from  measurements  which  they  have 
made  for  themselves. 

7.  Problems  in  mensuration  are  usually  made  clearer  by  dia- 
grams or  drawings  made  to  a  scale. 

THE  EIGHTH  GRADE 

The  work  of  the  eighth  grade  is  an  extension  of  the 
work  of  the  seventh  to  a  wider  range  of  applications  and 
it  takes  on  a  more  adult  point  of  view.  If  square  root 
is  to  be  taught  at  all,  it  is  taken  up  in  this  grade.  If 
the  work  of  arithmetic  is  given  in  order  that  the  pupil 
may  have  the  ability  to  interpret  the  quantitative  phases 


THE   COURSE   OF   STUDY   IN   ARITHMETIC     219 

of  his  environment,  it  would  seem  that  the  time  spent  in 
square  root  and  its  applications  might  better  be  spent 
with  other  problems. 

At  the  end  of  this  year  fundamental  principles  should 
be  thoroughly  understood,  habits  of  accuracy  fixed,  and 
readiness  and  a  fair  degree  of  speed  in  ordinary  computa- 
tions attained.  Power  to  state  a  problem  clearly,  to 
analyze  it  logically,  to  choose  a  good  method  of  solution, 
and  to  do  the  work  by  the  shortest  method  should  have 
been  acquired.  The  work,  then,  should  include : 

1.  A  general  application  of  the  work  of  the  preceding 
grades  to  problems  of  human  interest. 

2.  An  extension  of  percentage  to  insurance;    taxes; 
national    revenues;     trade    discount;     successive   trade 
discounts;    simple  interest;    bank  discount;    stock  in- 
vestments;   bond   investments. 

3.  An  extension  of  mensuration  to  pyramids,  cones, 
and  the  Pythagorean  Theorem.    Also  some  simple  geo- 
metric constructions  and  observations;  and  the  three 
forms  of  graphs  used  in  representing  statistics. 

4.  The  use  and  interpretation  of  formulae  as  "short- 
hand expressions"  of  the  principles  and  rules  of  mensura- 
tion. 

5.  The  use  of  ratio  and  proportion  in  expressing  the 
relation  of  similar  figures  and  in  the  problems  of  simple 
machines,  as  the  lever,  inclined  plane,  screw,  wedge,  etc. 

SUGGESTIONS.  —  1.  If  time  permits,  the  indirect  problem 
of  percentage  may  be  taken  up.  There  are  but  few  real  appli- 
cations of  it.  It  is  needed  to  find  a  selling  price  that  will  yield 
a  certain  per  cent  of  profit  upon  itself  when  the  cost  is  known ; 


220  THE   TEACHING  OF  ARITHMETIC 

also  to  find  a  list  price  that  may  be  discounted  and  yet  yield  a 
certain  profit  when  the  cost  is  known. 

2.  The  applications  of  percentage  are  difficult  to  teach  only 
because  the  transactions  are  outside  of  the  experiences  of  many 
of  the  pupils.  Seek  to  make  the  work  as  realistic  as  possible. 
Pupils  of  this  age  usually  like  to  play  at  "make-believe"  bill- 
ing clerks,  retail  merchants,  wholesalers,  bank  cashiers,  etc. 
This  is  done  in  all  commercial  schools. 

3.  Encourage  pupils  to  bring  in  all  kinds  of  commercial  pa- 
pers —  price  lists,  invoices  of  goods,  receipts,  checks,  promissory 
notes,  insurance  policies,  tax  bills,  etc.     As  these  are  collected 
by  the  pupils  they  should  be  carefully  preserved  and  kept  "in 
stock"  as  part  of  the  illustrative  material  of  the  school. 

4.  In  "stock  investments"  there  are  but  two  real  problems, 
viz. :   (1)  Stock  bought  at  one  price  and  sold  at  another  is  a  gain 
or  a  loss  to  the  buyer  of  how  much ;    (2)  stock  costing  a  certain 
price  and  paying  a  certain  dividend  makes  the  investor  what 
per  cent  of  the  investment.     These  problems  are  no  more  diffi- 
cult than  many  encountered  much  earlier  in  the  course.     The 
pupil's  difficulty  with  them  lies  in  the  fact  that  they  are  not 
concrete  to  him.    They  lie  entirely  without  his  range  of  experi- 
ences.    Seek  to  make  the  topic  realistic  by  a  study  of  some  local 
corporation,  by  the  issuing  of  stock  certificates  upon  "make- 
believe"  corporations  organized  by  the  pupils,  and  by  use  of 
problems  made  by  the  pupils  from  the  market  reports  of  the 
daily  papers. 

5.  The  study  of  bonds  may  be  made  more  real  in  connec- 
tion with  the  study  of  civil  government  in  considering  the  meth- 
ods used  by  municipalities,  states,  or  governments  of  raising 
money.     It  is  best  to  begin  with  a  study  of  your  own  city. 

6.  It  is  well  to  explain  the  functions  of  savings  banks,  banks 
of  deposit,  and  other  corporations.    A  visit  to  sjuch  places  is 
well  worth  while. 


CHAPTER  XVII 
MEASURING  RESULTS 

UNTIL  recently  the  rating  of  a  pupil  has  depended  upon 
the  personal  opinion  of  the  teacher  and  upon  set  examina- 
tions. During  the  last  few  years  considerable  work  has  been 
done  along  a  method  of  measuring  results,  known  as  "  stand- 
ard tests,"  that  promises  a  more  scientific  means  of  measur- 
ing the  products  of  certain  lines  of  school  training. 

USES  OF  STANDARDS  IN  MEASURING  EFFICIENCY 

There  are  at  least  three  general  uses  or  purposes  of 
standards  in  measuring  efficiency.  There  is  a  need  of  tests 
and  standards  to  be  used  in  the  administration  of  a  school 
system,  also,  for  determining  the  rating  of  an  individ- 
ual for  purposes  of  promotion ;  and,  finally,  tests  for  the 
purpose  of  analyzing  the  primary  defects  that  lead  to 
backwardness  in  any  subject.  These  may  be  classified 
as  administrative,  grading,  and  diagnostic  tests.  A 
standard  test  that  will  meet  one  of  these  three  purposes 
need  not  necessarily  meet  another.  Thus,  from  the 
standpoint  of  administration,  the  test  must  show  merely 
the  general  ranking  of  a  large  group,  or  of  an  entire  school 
system  in  a  finished  product ;  as,  for  example,  written  addi- 
tion. But  the  ability,  to  do  written  addition  is  a  complex 

221 


222  THE    TEACHING   OF   ARITHMETIC 

consisting  of  several  primary  abilities  with  which  the  ad- 
ministration is  not  directly  concerned. 

From  the  standpoint  of  grading  a  pupil  for  entrance 
or  promotion,  the  test  must  measure  the  individual  ability 
instead  of  group  ability,  and,  hence,  must  be  so  made  as 
to  eliminate  more  nearly  the  element  of  chance  error. 
Thus  the  average  score  from  a  single  example  in  addition 
given  to  several  thousand  pupils  might  mark  fairly  well 
the  general  ability  of  that  group  in  that  subject  and  thus 
answer  the  purposes  of  an  administrative  test ;  but  such 
a  test  would  be  entirely  unreliable  for  a  single  individual, 
owing  to  the  element  of  chance.  A  grading  test,  moreover, 
must  be  a  test  of  the  ability  of  the  individual  in  a  finished 
product,  not  of  abilities  needed  in  that  finished  product. 

The  diagnostic  tests  are  tests  given  to  discover  causes  of 
backwardness  in  performing  any  finished  process.  They 
must  locate  the  particular  primary  ability  or  abilities  that 
need  special  training.  The  weakness  in  any  final  process 
may  be  due  to  a  weakness  in  one  or  more  of  several  prac- 
tically unrelated  primary  abilities.  Thus  failure  in  the 
finished  product  of  written  addition  may  be  due  to  the 
lack  of  any  one  of  the  following  primary  abilities :  (1)  the 
automatic  control  of  the  primary  number  facts ;  (2)  abil- 
ity to  hold  in  mind  a  two-  or  a  three-figured  number  and 
add  it  to  a  one-figured  number;  (3)  ability  to  carry  in 
mind  the  "tens"  of  each  column- sum  and  add  them  to 
the  next  column;  and  (4)  ability  to  record  accurately 
the  "ones"  figure  of  each  column-sum.  Ability  in  any 
one  of  these  does  not  imply  ability  in  any  other,  so  diagnos- 
tic tests  to  show  the  source  of  weakness  must  be  made 


MEASURING   RESULTS  223 

before  corrective  drill  work  can  be  wisely  applied.  Thus, 
if  the  weakness  in  final  addition  is  due  to  (2),  outlined 
above,  as  is  often  the  case,  drill  upon  (1)  is  of  almost  no 
help,  yet  this  is  often  the  only  source  of  training  prepara- 
tory to  written  work.  Thus  it  is  seen  that  tests  to  be  of 
any  real  service  in  supervising,  promoting,  or  correcting 
weaknesses  must  be  worked  out  carefully  according  to 
the  purposes  that  they  are  to  serve. 

ATTEMPTED  SOLUTIONS  OF  THE  PROBLEM 

While  many  persons  have  worked  at  the  problems  out- 
lined above,  usually  the  work  has  been  restricted  to  special 
classes  or  school  systems.  Of  those  who  have  worked  in  a 
wider  field,  the  work  of  Rice,  Stone,  Courtis,  and  Woody 
is  best  known.  The  real  problem  of  Rice  and  Stone, 
however,  was  not  that  of  establishing  a  standard  of  meas- 
urement, but  since  then*  questions  and  results  are  used 
by  school  systems  in  comparing  their  own  schools  with 
those  examined  by  these  men,  their  questions  are  often 
classed  among  the  "standard  tests." 

THE  WORK  OF  RICE 

The  tests  given  by  Mr.  J.  M.  Rice  1  in  1902  were  not 
intended  to  become  standards.  They  consisted  of  five 
sets  of  questions  of  eight  each,  for  grades  four  to  eight, 
inclusive,  and  were  given  to  the  pupils  of  seven  city  sys- 
tems, aggregating  over  6000  pupils,  in  order  to  determine 
some  of  the  factors  upon  which  successful  school  work 
depends.  They  are  of  interest  in  showing  the  beginning 
i  The  Forum,  XXXIV  (October-December),  1902. 


224  THE   TEACHING  OF  ARITHMETIC 

of  the  attempts  that  later  led  to  more  scientific  attempts 
at  standardization. 

THE  QUESTIONS  USED  BY  RICE 
FOURTH  YEAR 

1.  A  man  bought  a  lot  of  land  for  $1743,  and  built  upon  it  a 
house  costing  $5482.     He  sold  them  both  for  $10,000.     How 
much  money  did  he  make? 

2.  If  a  boy  pays  $2.83  for  a  hundred  papers,  and  sells  them  at 
4  cents  apiece,  how  much  money  does  he  make? 

3.  If  there  were  4839  classrooms  in  New  York  City,  and  47 
children  in  each  classroom,  how  many  children  would  there  be  in 
the  New  York  schools  ? 

4.  A  man  bought  a  farm  for  $16,575,  paying  $85  an  acre. 
How  many  acres  were  there  in  the  farm? 

5.  What  will  24  quarts  of  cream  cost  at  $1.20  a  gallon  ? 

6.  A  lady  bought  4  pounds  of  coffee  at  27  cents  a  pound,  16 
pounds  of  flour  at  4  cents  a  pound,  15  pounds  of  sugar  at  6  cents  a 
pound,  and  a  basket  of  peaches  for  95  cents.     She  handed  the 
storekeeper  a  $10  note.     How  much  change  did  she  receive? 

7.  I  have  $9786.     How  much  more  must  I  have  in  order  to  be 
able  to  pay  for  a  farm  worth  $17,225  ? 

8.  If  I  buy  8  dozen  pencils  at  37  cents  a  dozen,  and  sell  them 
at  5  cents  apiece,  how  much  money  do  I  make  ? 

FIFTH  YEAR 

1.  A  man  bought  a  lot  of  land  for  $1743,  and  built  upon  it  a 
house  costing  $5482.     He  sold  them  both  together  for  $10,000. 
How  much  did  he  make  ? 

2.  If  a  boy  pays  $2.83  for  a  hundred  papers,  and  sells  them  at 
4  cents  apiece,  how  much  does  he  make? 

3.  What  will  24  quarts  of  cream  cost  at  $1.20  a  gallon? 

4.  If  I  buy  8  dozen  pencils  at  37  cents  a  dozen,  and  sell  them 
at  5  cents  apiece,  how  much  money  do  I  make  ? 

5.  A  flour  merchant  bought  1437  barrels  of  flour  at  $7  a  barrel. 
He  sold  900  of  these  barrels  at  $9  a  barrel,  and  the  remainder  at 
$6  a  barrel.    How  much  did  he  make? 


MEASURING  RESULTS  225 

6.  How  many  feet  long  is  a  telegraph  wire  extending  from  New 
York  to  New  Haven,  a  distance  of  74  miles?     There  are  5280 
feet  in  a  mile. 

7.  A  merchant  bought  15  pieces  of  cloth,  each  containing  62 
yards.     He  sold  234  yards.     How  many  dress  patterns  of  12 
yards  each  did  he  have  left? 

8.  Frank  had  $3.08.     He  spent  J  of  it  for  a  cap,  \  of  it  for  a 
ball,  and  with  the  remainder  bought  a  book.     How  much  did 
the  book  cost  ? 

SIXTH  YEAR 

1.  If  a  boy  pays  $2.83  for  a  hundred  papers,  and  sells  them  at 
4  cents  apiece,  how  much  does  he  make? 

2.  What  will  24  quarts  of  cream  cost  at  $1.20  a  gallon? 

3.  If  I  buy  8  dozen  pencils  at  37  cents  a  dozen,  and  sell  them 
at  5  cents  apiece,  how  much  do  I  make? 

4.  A  flour  merchant  bought  1437  barrels  of  flour  at  $7  a  barrel. 
He  sold  900  of  these  barrels  at  $9  a  barrel,  and  the  remainder  at 
$6  a  barrel.     How  much  did  he  make  ? 

5.  If  a  train  runs  31$  miles  an  hour,  how  long  will  it  take  the 
train  to  run  from  Buffalo  to  Omaha,  a  distance  of  1045  miles  ? 

6.  If  a  map  10  inches  wide  and  16  inches  long  is  made  on  a 
scale  of  50  miles  to  the  inch,  what  is  the  area  in  square  miles  that 
the  map  represents? 

7.  The  salt  water  which  was  obtained  from  the  bottom  of  a 
mine  of  rock  salt  contained  0.08  of  its  weight  of  pure  salt.     What 
weight  of  salt  water  was  it  necessary  to  evaporate  in  order  to  ob- 
tain 3896  pounds  of  salt? 

8.  A  gentleman  gave  away  \  of  the  books  in  his  library,  lent 
J  of  the  remainder,  and  sold  \  of  what  was  left.     He  then  had  420 
books  remaining.     How  many  had  he  at  first  ? 

SEVENTH  YEAR 

1.  If  a  map  10  inches  wide  and  16  inches  long  is  made  on  a 
scale  of  50  miles  to  the  inch,  what  is  the  area  in  square  miles  that 
the  map  represents? 

2.  The  salt  water  which  was  obtained  from  the  bottom  of  a 
mine  of  rock  salt  contained  0.08  of  its  weight  of  pure  salt.     What 


226  THE   TEACHING  OF   ARITHMETIC 

weight  of  salt  water  was  it  necessary  to  evaporate  in  order  to  ob- 
tain 3896  pounds  of  salt  ?      \3 1f  '<*  * 

3.  A  gentleman  gave  away  |  of  the  books  in  his  library,  lent 
\  of  the  remainder,  and  sold  \  of  what  was  left.     He  then  had  420 
books  remaining.     How  many  had  he  at  first  ? 

4.  A  farmer's  wife  bought  2.75  yards  of  table  linen  at  $0.87  a 
yard  and  16  yards  of  flannel  at  $0.55  a  yard.     She  paid  in  butter 
at  $0.27  a  pound.     How  many  pounds  of  butter  was  she  obliged 
to  give?          ^7^ 

5.  If  coffee  sold  at  33  cents  a  pound  gives  a  profit  of  10  per 
cent,  what  per  cent  of  profit  would  there  be  if  it  were  sold  at  36 
cents  a  pound  ? 

6.  Sold  steel  at  $27.60  a  ton,  with  a  profit  of  15  per  cent,  and  a 
total  profit  of  $184.50.     What  quantity  was  sold  ? 

7.  If  a  woman  can  weave  1  inch  of  rag  carpet  a  yard  wide  in 
4  minutes,  how  many  hours  will  she  be  obliged  to  work  in  order  to 
weave  the  carpet  for  a  room  24  feet  long  and  24  feet  wide  ?    No 
deduction  to  be  made  for  waste.    ^3*f*7  ^" 

8.  A  fruit  dealer  bought  300  apples  at  the  rate  of  5  for  a  cent, 
and  300  at  4  for  a  cent.     He  sold  them  all  at  the  rate  of  ,8  for  5 
cents.     What  per  cent  did  he  gain  on  his  investment  ? 

EIGHTH  YEAB 

1.  If  a  map  10  inches  wide  and  16  inches  long  is  made  on  a 
scale  of  50  miles  to  the  inch,  what  is  the  area  in  square  miles  that 
the  map  represents? 

2.  The  salt  water  which  was  obtained  from  the  bottom  of  a 
mine  of  rock  salt  contained  0.08  of  its  weight  of  pure  salt.     What 
weight  of  salt  water  was  it  necessary  to  evaporate  in  order  to  ob- 
tain 3896  pounds  of  salt? 

3.  A  gentleman  gave  away  f  of  the  books  in  his  library,  lent  \ 
of  the  remainder,  and  sold  \  of  what  was  left.     He  then  had  420 
books  remaining.     How  many  had  he  at  first  ? 

/4.  A  man  sold  50  horses  at  $126.00  each.  On  one  half  of  them 
he  made  20  per  cent,  and  on  the  other  half  he  lost  10  per  cent. 
How  much  did  he  gain  ? 

5.  Sold  steel  at  $27.60  a  ton,  with  a  profit  of  15  per  cent,  and  a 
total  profit  of  $184.50.     What  quantity  was  sold? 

6.  A  fruit  dealer  bought  300  apples  at  the  rate  of  5  for  a  cent, 


MEASURING   RESULTS 


227 


and  300  at  4  for  a  cent.     He  sold  them  all  at  the  rate  of  8  for  5 
cents.     What  per  cent  did  he  gain  on  his  investment  ? 

7.  The  insurance  on  f  of  the  value  of  a  hotel  and  furniture  cost 
$420.00.     The  rate  being  70  cents  on  $100.00,  what  was  the  value 
of  the  property? 

8.  Gunpowder  is  composed  of  niter  15  parts,  charcoal  3  parts, 
and  sulphur  2  parts.     How  much  of  each  in  360  pounds  of 
powder  ?  , 

RESULTS  FOUND  BY  RICE 

The  results  of  the  tests  and  many  interesting  conclusions 
derived  from  them  can  be  found  in  an  article  by  Mr.  Rice 
in  The  Forum  for  October-December,  1902.  The  follow- 
ing chart  shows  the  averages  for  the  schools  in  each  of  the 
seven  cities. 


/•"I,—  — 

GRADE 

GRADE 

GRADE 

GRADE 

GRADE 

SCHOOL 

Vj/ITT 

IV 

V 

VI 

VII  ,. 

VIII 

AVERAGE 

III 

68.4 

79.5 

79.3 

81.1 

91.7 

80.0 

I 

72.7 

84.7 

80.4 

64.2 

80.9 

76.6 

I 



80.3 

80.9 

43.5 

72.7 

69.3 

I 

54.5 

74.7 

72.2 

63.5 

74.5 

67.8 

I 

60.0 

70.8 

69.6 

54.6 

66.5 

64.3 

II 

81.3 

78.2 

71.2 

33.6 

36.8 

60.2 

III 

70.1 

53.6 

43.7 

53.9 

51.1 

54.5 

IV 

70.5 

73.2 

58.9 

31.2 

41.6 

55.1 

IV 

62.9 

70.5 

59.8 



22.5 

.  53.9 

IV 

53.5 

53.5 

42.3 

16.1 

48.7 

42.8 

IV 

59.8 

65.3 

54.9 

35.2 

43.5 

51.5 

V 

38.5 

67.0 

44.1 

29.2 

51.1 

45.9 

VI 

28.1 

38.1 

68.3 

33.5 

26.9 

39.0 

VI 

41.6 

45.3 

46.1 

19.5 

30.2 

36.5 

VI 

36.8 

55.0 

34.5 

30.5 

23.3 

36.0 

VII 

59.3 

53.7 

35.2 

29.1 

25.1 

40.5 

VII 

47.4 

65.4 

35.2 

15.0 

19.6 

36.5 

VII 

41.1 

37.5 

27.6 

8.9 

11.3 

25.3 

Gen.  Av. 

59.5 

69.4 

60.7 

39.4 

49.4 

55.7 

228  THE   TEACHING  OF  ARITHMETIC 

THE  WORK  OF  STONE 

Dr.  C.  W.  Stone,1  in  1908,  made  a  study  of  "Arithmetical 
Abilities  and  Some  of  the  Factors  Determining  Them." 
This  was  a  study  of  the  nature  of  the  product  of  the  first 
six  years'  work  in  arithmetic.  The  material  gathered  was 
over  6000  test  papers  from  152  classrooms  hi  26  different 
school  systems.  The  questions  and  results  have  been  so 
widely  used  that  they  are  sometimes  known  as  "Stone's 
Standard  Tests."  A  study  of  them  will  show  that  they  do 
not  meet  the  needs  of  a  standard  test  for  any  of  the  three 
uses  outlined  in  the  first  part  of  this  chapter,  nor  were  they 
so  intended  by  Dr.  Stone.  They  have  their  value,  how- 
ever, and  are  reproduced  here  on  account  of  the  general 
interest  in  them.  Pupils  were  allowed  12  minutes  for  the 
test  on  fundamental  operations  and  15  minutes  for  the 
test  on  reasoning. 

THE  STONE  TESTS 
TEST  IN  FUNDAMENTAL  OPERATIONS 

Work  as  many  of  these  problems  as  you  have  time  for ;  work 
them  in  order  as  numbered. 

1.  Add          2375 

4052 
6354 
260 
6041 
1543 

1  C.  W.  Stone,  Arithmetical  Abilities  and  Some  Factors 
Determining  Them,  Published  by  Teachers  College,  Columbia 
University,  New  York  City,  1908. 


MEASURING  RESULTS  229 

2.  Multiply  3265  by  20. 

3.  Divide  3328  by  64. 

4.  Add  596 

428 

94 

75 
302 
645 
984 
897 

5.  Multiply  768  by  604. 

6.  Divide  1918962  by  543. 

7.  Add          4695 

872 
7948 
6786 

567 

858 
9447 
7499 

8.  Multiply  976  by  87. 

9.  Divide  2782542  by  679. 

10.  Multiply  5489  by  9876. 

11.  Divide  5099941  by  749. 

12.  Multiply  876  by  79. 

13.  Divide  62693256  by  859. 

14.  Multiply  96879  by  896. 

TEST  IN  REASONING 

Solve  as  many  of  the  following  problems  as  you  have  time  for ; 
work  them  in  order  as  numbered : 

1.  If  you  buy  2  tablets  at  7  cents  each  and  a  book  for  65  cents, 
how  much  change  should  you  receive  from  a  two-dollar  bill  ? 


230  THE   TEACHING  OF   ARITHMETIC 

2.  John  sold  4  Saturday  Evening  Posts  at  5  cents  each.     He 
kept  %  the  money  and  with  the  other  £  he  bought  Sunday  papers 
at  2  cents  each.    How  many  did  he  buy  ? 

3.  If  James  had  4  times  as  much  money  as  George,  he  would 
have  $16.    How  much  money  has  George? 

4.  How  many  pencils  can  you  buy  for  50  cents  at  the  rate  of 
2  for  5  cents? 

5.  The  uniforms  for  a  baseball  nine  cost  $2.50  each.     The 
shoes  cost  $2  a  pair.     What  was  the  total  cost  of  uniforms  and 
shoes  for  the  nine  ? 

6.  In  the  schools  of  a  certain  city  there  are  2,200  pupils ;  |  are 
in  the  primary  grades,  J  in  the  grammar  grades,  f  in  the  High 
School  and  the  rest  in  the  night  school.     How  many  pupils  are 
there  in  the  night  school? 

7.  If  85  tons  of  coal  cost  $21,  what  will  5§  tons  cost? 

8.  A  news  dealer  bought  some  magazines  for  $1.     He  sold 
them  for  $1.20,  gaining  5  cents  on  each  magazine.     How  many 
magazines  were  there? 

9.  A  girl  spent  |  of  her  money  for  car  fare,  and  three  times  as 
much  for  clothes.     Half  of  what  she  had  left  was  80  cents.     How 
much  money  did  she  have  at  first  ? 

10.  Two  girls  receive  $2.10  for  making  button-holes.     One 
makes  42,  the  other  28.     How  shall  they  divide  the  money  ? 

11.  Mr.  Brown  paid  one  third  of  the  cost  of  a  building ;  Mr. 
Johnson  paid  §  the  cost.     Mr.  Johnson  received  $500  more  annual 
rent  than  Mr.  Brown.     How  much  did  each  receive? 

12.  A  freight  train  left  Albany  for  New  York  at  6  o'clock. 
An  express  left  on  the  same  track  at  8  o'clock.      It  went  at 
the  rate  of  40  miles  an  hour.      At  what  time  of  day  will  it 
overtake  the  freight  train  if  the  freight  train  stops  after  it  has 
gone  56  miles  ? 

PERCENTAGE  OF  MISTAKES 

Stone  published  38  tables  from  the  scores  obtained. 
Those  of  most  interest  to  teachers  and  supervisors  of  arith- 
metic are  Tables  VIII  and  IX,  showing  errors  in  reasoning 
and  in  addition. 


MEASURING   RESULTS  231 

TABLE  VIII  TABLE  IX 


MISTAKES  IN  REASONING 

MISTAKES  IN  ADDITION 

'     -a>s|s 

•ill 

oa?-*3 

QJT5   M   » 

fg&l 

£ 

li 

•-  a 

& 

No.  of 
problems 
incorrect 

No.  of 
problems 
attempted 

Systems  in 
order  of 
per  cent  of 
mistakes 

Per  cent 
incorrect 

1 

«£ 

"8  11 

o'ffl-S 
Z 

No.  of 
steps 
attempted 

XVI 

45.1 

359 

796 

XXII  

14.5 

196 

634 

XVII  

44.9 

335 

746 

XX  

13.5 

139 

595 

XXIII  

41.1 

238 

579 

I  

13.4 

115 

861 

Ill  

36.7 

282 

769 

XVII  

10.5 

96 

918 

I.  _  

34.7 

269 

776 

XVIII  

10.4 

102 

982 

XV  

33.7 

249 

739 

VI  

10.1 

91 

908 

VI  

31.8 

233 

733 

X  

9.9 

81 

819 

XXII  

30.9 

196 

634 

IX  

9.6 

86 

898 

X...  

29.7 

232 

781 

V  

9.2 

89 

967 

XXIV.  

28.9 

167 

577 

XIII  

8.9 

75 

847 

XIII  

28.8 

230 

799 

XV  

8.8 

72 

818 

XXVI  

28.6 

276 

964 

VII  

8.76 

87 

993 

VIII  

27.9 

192 

689 

IV  

8.5 

81 

951 

XVIII  

27 

175 

648 

XXV  

8.3 

61 

739 

XIX  

26.4 

255 

965 

XXIII  

8 

56 

703 

XXV  

25.3 

150 

592 

VIII._  _  

7.5 

60 

803 

IV  

25.1 

147 

585 

XVI.  _  

7.04 

67 

952 

XX  

23.4 

139 

595 

II  

7 

60 

857 

VII  

23.1 

189 

819 

Ill  

6.5 

55 

843 

XIV  

22.9 

175 

765 

XII  

6.3 

56 

896 

XII  

22.3 

185 

831 

XIV.__  

6.2 

57 

921 

IX  

20.3 

161 

794 

XXIV  

5.9 

54 

918 

II  

19.7 

137 

696 

XXVI  

5.8 

54 

930 

V..  

18.6 

171 

919 

XIX  

5.78 

57 

987 

XXI  

15.7 

106 

674 

XXI  

5.1 

44 

860 

XL. 

14.4 

112 

776 

XI 

4.7 

42 

888 

232  THE   TEACHING  OF  ARITHMETIC 

THE  WORK  or  COURTIS 

Mr.  S.  A.  Courtis,1  then  head  of  the  department  of 
mathematics  in  the  Detroit  Home  and  Day  School, 
through  the  inspiration  received  from  the  work  of  Stone, 
began  his  investigations  in  1909.  From  the  first,  his 
purpose  was  to  develop  a  series  of  standard  tests  to  be 
used  in  measuring  the  results  of  instruction  in  arithmetic, 
and  he  is  the  first  to  take  up  this  particular  problem.  The 
first  of  the  " Courtis  tests"  consists  of  eight  tests  known  as 
"Series  A."  The  first  published  standards  were  derived 
from  the  average  results  of  testing  a  total  of  about  9000 
pupils  in  all  grades  from  the  third  to  the  eighth  inclusive. 
His  later  standards  for  the  same  series  are  made  up  from 
the  results  obtained  from  67,000  pupils  distributed  pretty 
generally  throughout  the  country.  The  standards  are 
not  the  average  of  conditions  as  they  were  found,  but  the 
estimated  scores  that  should  exist  at  the  end  of  the  year. 
"  Series  A,"  however,  was  full  of  defects.  While  more 
diagnostic  than  administrative,  it  was  neither,  yet  it 
was  evidently  intended  for  the  latter,  for  Courtis  says, 
"The  general  purpose  of  testing  work  is  not  to  measure 
the  abilities  of  individuals  to  determine  their  fitness 
for  promotion,  but  to  reveal  the  efficiency  of  school 
procedure." 

But  Courtis  was  not  blinded  by  enthusiasm  to  the  de- 
fects of  his  first  work.  In  1913  he  published  a  new  set  of 
tests  and  standards  known  as  "Series  B"  which  greatly 

1  Courtis  Standard  Research  Tests,  published  by  Department  of 
Cooperative  Research,  82  Eliot  Street,  Detroit,  Michigan. 


MEASURING  RESULTS  233 

eliminated  the  defects  of  "Series  A"  from  the  standpoint 
of  its  purpose  as  expressed  in  the  above  quotation. 

This  new  series  consists  of  but  four  tests,  each  consisting 
of  but  a  single  written  process,  and  the  scores  are  median l 
scores  rather  than  average  scores.  With  the  exception 
that  Courtis  has  made  but  one  test  in  each  process,  to  be 
used  in  all  grades  from  the  third  to  the  eighth  inclusive, 
instead  of  a  series  for  primary  grades  and  another  for 
grammar  grades  with  a  known  relation  between  them, 
he  has  a  series  exceptionally  well  suited  to  measure  the 
abilities  in  the  four  fundamental  processes.  Test  one  is 
an  eight-minute  test  in  written  addition  and  consists  of 
twenty-four  exercises  of  equal  weight,  each  being  made  of 
nine  three-figured  numbers.  Such  a  test  allows  enough 
time  devoted  to  a  single  process  to  eliminate  more  nearly 
the  accidental  errors  and  thus  it  becomes  a  much  more 
reliable  measure  than  does  a  test  of  mixed  exercises  given 
for  a  shorter  period. 

Test  two  is  a  four-minute  test  in  written  subtraction 
consisting  of  twenty-four  exercises  with  eight-figured 
numbers.  Test  three  is  a  six-minute  test  in  multiplica- 
tion, and  test  four  an  eight-minute  test  in  division. 

There  are  four  forms  of  the  same  weighting  in  each  test. 
One  form  of  each  is  given  here. 

1  The  median  score  is  found  by  arranging  the  scores  in  order 
of  magnitude  from  the  best  down  to  the  poorest  and  taking  the 
middle  case.  Thus  of  35  scores  so  arranged  the  18th  is  the 
median,  for  there  are  17  poorer  and  17  better.  It  differs  slightly 
from  the  average,  for  each  score  influences  the  median  only  as  a 
single  case,  whereas,  in  calculating  an  average,  very  small  or  very 
large  scores  have  a  very  marked  effect  upon  the  general  average. 


234  THE   TEACHING  OF   ARITHMETIC 

COURTIS   STANDARD   RESEARCH   TESTS 


Arithmetic    Test  No.  1    Addition 
Series  B  Form  1 


SCORE 

No.  Attempted.. 
No.  Right... 


You  will  be  given  eight  minutes  to  find  the  answers  to  as  many 
of  these  addition  examples  as  possible.  Write  the  answers  on  this 
paper  directly  underneath  the  examples.  You  are  not  expected 

to  be  able  to  do  them  all.  You  will  be  marked  for  both  speed  and 

accuracy,  but  it  is  more  important  to  have  your  answers  right 
than  to  try  a  great  many  examples. 

927    297    136  486    384  176  277  837 

379    925    340  765    477  783  445  882 

756    473    988  624    881  697  682  959 

837  983    386  140    266  200  594  603 
924    315    353  812    679  366  481  118 
110    661    904  466    241  851  778  781 
854    794    547  355    796  535  849  756 
965    177    192  834    850  323  157  222 
344    124    439  667    733  229  953  626 

637    664    634  672    226  351  428  862 

695    278    168  253    880  788  975  159 

471    345    717  948    663  705  450  383 

913    921    142  629    819  174  194  451 

564    787    449  936    779  426  666  938 

932    646    453  223    123  649  742  433 

559    433    924  358    338  755  295  599 

106    464    659  676    996  140  187  172 

228    449    432  122    303  246  281  152 

677    223    186  275    432  634  647  588 

464    878    478  521    876  327  197  256 

234    682    927  854    571  327  685  719 

718    399    516  939    917  394  678  624 

838  904    923  582    749  807  456  969 
293    353    653  566    495  169  393  761 
423    419    216  936    250  491  625  113 
955    756    669  472    833  885  240  449 
619    314    409  264    318  403  152  122 


MEASURING   RESULTS  235 

COURTIS    STANDARD    RESEARCH   TESTS 


Arithmetic     Test  No.  2     Subtraction 
Series  B  Form  1 


SCORE 

No.  Attempted. 
No.  Right.. 


You  will  be  given  four  minutes  to  find  the  answers  to  as  many 
of  these  subtraction  examples  as  possible.  Write  the  answers  on 
this  paper  directly  underneath  the  examples.  You  are  not  ex- 
pected to  be  able  to  do  them  all.  You  will  be  marked  for  both 
speed  and  accuracy,  but  it  is  more  important  to  have  your  answers 
right  than  to  try  a  great  many  examples. 


107796491 
77197029 


75088824 
57406394 


91500053 
19901563 


87939983 
72207316 


160620971 
80361837 


51274387 
25842708 


117359208 
36955523 


47222970 
17504943 


115364741 
80195261 


67298125 
29346861 


92057352 
42689037 


113380936 
42556840 


64547329 
48813139 


121961783 
90492726 


109514632 
81268615 


125778972 
30393060 


92971900 
62207032 


104339409 
74835938 


60472960 
50196521 


119811864 
34379846 


137769153 
70176835 


144694835 
74199225 


123822790 
40568814 


80836465 
49178036 


Name.. 

School. 
City 


, ,  Age  last  birthday. 

BOT   OB   GIRL 

Grade Room....  j ... 

.State....  ....Date.... 


236  THE   TEACHING  OF  ARITHMETIC 

COURTIS   STANDARD   RESEARCH   TESTS 


Arithmetic     Test  No.  3     Multiplication 
Series  B  Form  1 


SCORE 

No.  Attempted. 
No.  Right.  _. 


You  will  be  given  six  minutes  to  work  as  many  of  these  multi- 
plication examples  as  possible.  You  are  not  expected  to  be  able 
to  do  them  all.  Do  your  work  directly  on  this  paper ;  use  no 
other.  You  will  be  marked  for  both  speed  and  accuracy,  but  it 
is  more  important  to  have  your  answers  right  than  to  try  a  great 
many  examples. 

8246  3597  5739  2648  9537 

29  73  86  46  92 


4268  7593  6428  8563  2947 

37  640  68  207  63 


5368  4792  7942  3586  9742 

95  84  72  36  69 


6385  8736  5942  6837  4952 

48  602  39  680  47 


3876  9245  7368  2594  6495 

93  86  74  25  19 


Name ,     ,  Age  last  birthday. 

BOT  OB  GIRL 

School....  Grade....  ....Room....   ;... 


City State. _ Date. 


MEASURING  RESULTS  237 

COURTIS   STANDARD   RESEARCH  TESTS 


Arithmetic.     Test  No.  4.     Division 
Series  B  Form  1 


SCORE 

No.  Attempted.. 
No.  Right.  _. 


You  will  be  given  eight  minutes  to  work  as  many  of  these  divi- 
sion examples  as  possible.  You  are  not  expected  to  be  able  to 
do  them  all.  Do  your  work  directly  on  this  paper ;  use  no  other. 
You  will  be  marked  for  both  speed  and  accuracy,  but  it  is  more 
important  to  have  your  answers  right  than  to  try  a  great  many 
examples. 


25)6776 


94)85352 


37)9990 


86)80066 


73)68765 


49)31409 


68)43520 


52)44252 


37)14467  86)60372  94)67774  26)9750 


68)39508  49)28420  52)21112  73)33653 


45)33796 


76)67000 


93)28458 


82)29602 


Name ,     Age  last  birthday. 

BOY  OB  GIRL 

School Grade Room....   j 

City.... State Date 


238  THE    TEACHING  OF   ARITHMETIC 


STANDARD  SCORES l 

In  the  table  below  will  be  found  median  speeds  and  accu- 
racies based  upon  distribution  of  many  thousands  of  individual 
scores  in  tests  given  in  May  or  June,  1915-1916.  The  distribu- 
tion for  each  grade  was  made  up  of  approximately  equal  numbers 
of  classes  from  large-city  schools  and  from  small-city  and  county 
schools.  One  or  two  of  the  medians  have  been  adjusted  slightly 
that  the  results  as  a  whole  might  yield  smooth  curves.  Half 
year  divisions  have  been  combined  to  make  whole  grades.  Com- 
parison of  these  results  with  those  in  the  tables  that  follow  will 
show  the  relation  of  the  median  scores,  and  of  the  values  adopted 
as  standards,  to  the  scores  from  tests  in  various  cities. 

TABLE 


M. 

s. 

M. 

8. 

M. 

S. 

M. 

S. 

3 

6.3 

4 

41 

100 

5.6 

5 

49 

100 

4 

7.4 

6 

64 

100 

7.4 

7 

80 

100 

5 

8.6 

8 

70 

100 

9.0 

9 

83 

100 

6 

9.8 

10 

73 

100 

10.3 

11 

85 

100 

7 

10.9 

11 

75 

100 

11.6 

12 

86 

100 

8        11.6        12        76        100  12.9  13  87        100 

M.           S.          M.            S.  M.  S.  M.           S. 

3  .8         0        .6  0       

4  6.2          6        67        100  4.6  4  57        100 

5  7.5          8        75        100  6.1  6  77        100 

6  9.1          9        78        100  8.2  8  87        100 

7  10.2        10        80        100  9.6  10  90        100 

8  11.5        11        81        100  10.7  11  91        100 

M  =  Adjusted  Median. 

S    =  Scores  adopted  as  standard. 

1  From  Bulletin  Number  Four,   Courtis  Standard   Research 
Tests. 


MEASURING   RESULTS  239 

FURTHER  COMMENTS  UPON  SERIES  B 

The  attempt  is  to  give  exercises  of  the  same  weight. 
An  examination  of  the  exercises  would  seem  to  indicate 
that  this  has  been  well  done  except  in  the  case  of  division. 
A  glance  at  the  answers  and  divisors  shows  a  much  greater 
variation  in  weight  than  do  the  other  three  tests.  For 
in  division  the  difficulty  does  not  depend  upon  the  number 
of  figures  involved,  but  upon  the  character  of  the  divisors 
and  quotients,  since  estimating  the  quotient  figures  con- 
stitutes the  greatest  difficulty.  » 

The  results  show  that  a  7th  grade  pupil  can  try 
about  11  or  12  exercises  in  both  addition  and  subtrac- 
tion, and  about  9  or  10  in  both  multiplication  and  division. 
This  eliminates  more  of  the  chance  errors  of  the  individual 
and  gives  a  more  reliable  result  than  those  obtained  in 
Series  A.  For  this  reason  they  become  not  only  a  measure 
of  the  efficiency  of  an  entire  school,  for  which  they  were 
intended,  but  a  rather  reliable  measure  of  the  abilities  of 
the  individual  for  promotional  purposes.  Yet  a  single 
attempt  will  not  furnish  a  very  safe  measure  of  the  indi- 
vidual, owing  to  the  chance  of  accidental  errors. 

FURTHER  COMMENTS  ON  SERIES  A 

When  Series  A  first  appeared,  the  novelty  of  measur- 
ing any  type  of  school  efficiency  by  a  fixed  standard 
awakened  great  enthusiasm  among  school  men  through- 
out the  country  and  no  doubt  led  to  a  great  many  con- 
clusions not  warranted  by  the  tests,  and  to  undue  emphasis 
upon  drills  to  develop  special  abilities  rather  than  upon  the 


240  THE   TEACHING  OF  ARITHMETIC 

completed  process.  Not  only  that,  but  in  many  cases  no 
doubt  undue  emphasis  was  placed  upon  drill  in  compu- 
tation to  the  neglect  of  the  applicative  side  of  the  subject 
to  the  solution  of  problems.  It  is  well,  then,  to  examine 
the  first  series,  which  are  too  well  known  to  be  repeated 
here,  to  see  what  each  of  them  really  tests  in  order  that 
teachers  may  examine  more  critically  other  tests  that 
appear. 

The  first  four  tests  are  "  speed  tests  "  in  recording  the 
primary  number  combinations  of  addition,  subtraction, 
multiplication,  and  division.  It  is  a  question  whether 
these  tests  test  the  knowledge  of  the  fundamental  number 
facts  as  much  as  they  do  the  motor  dexterity  of  the  pupil 
in  recording  the  fact.  That  is,  a  high  score  need  not  neces- 
sarily show  superior  control  of  the  facts,  nor  a  low  score 
the  lack  of  such  a  control,  but  the  score  may  be  condi- 
tioned by  the  pupil's  ability  to  make  figures.  But  even  if 
they  do  measure  the  control  of  the  primary  number  facts, 
they  are  diagnostic  tests  needed  in  the  analysis  of  the 
cause  of  backwardness,  and  not  administrative  tests  to 
determine  the  efficiency  of  school  instruction.  The 
administration  is  chiefly  interested  in  the  finished  process. 

The  tests,  however,  are  not  suited  for  diagnostic  tests  — 
a  thing  for  which  they  were  not  intended  —  for  the  stand- 
ards or  scores  are  the  work  of  but  1^  minutes  upon  each 
test.  Manifestly  the  work  of  an  individual  for  1^  minutes 
cannot  be  taken  as  a  reliable  estimate  of  his  ability.  For 
all  who  have  worked  with  children  know  how  largely  the 
element  of  chance  enters  into  the  work  of  so  brief  a 
period. 


MEASURING  RESULTS  241 

The  first  four  tests,  then,  as  they  now  stand,  are  not 
tests  for  measuring  the  abilities  of  the  individual  either  to 
determine  his  fitness  for  promotion  or  to  discover  weakness 
that  must  be  corrected  in  order  to  secure  efficient  work  in 
the  written  processes,  and  they  should  not  be  used  for 
such  purposes  by  the  teacher  or  supervisor. 

Courtis  in  Bulletin  Number  Four  says :  "  In  June,  1912, 
a  set  of  tests  of  the  four  fundamental  processes  in  arithme- 
tic was  issued,  because  investigations  then  completed  had 
proved  that  the  tests  of  Series  A  were  of  little  practical  value 
except  as  instruments  of  research." 

Test  five  is  a  speed  test  in  copying  figures  and  thus  tests 
but  motor  control.  It  is  thus  more  elementary  than  any 
of  the  first  four  and  belongs  even  more  nearly  to  the  diag- 
nostic class  of  tests  than  they  do.  As  an  administrative 
test  it  is  of  but  little  use,  but  it  is  a  very  necessary  diag- 
nostic test  for  determining  the  cause  of  slow  work  in  any 
written  process. 

Test  number  six  is  called  "a  speed  test  in  reasoning." 
In  attempting  to  cover  so  wide  a  range  as  from  the  third 
to  the  eighth  grades  inclusive,  the  results  cannot  be  of 
great  value.  One's  ability  to  discover  what  to  do  in  a 
problem  depends  upon  his  ability  to  read  understandingly, 
upon  his  experiences  through  which  he  can  interpret  the 
situation  described,  and,  finally,  upon  his  knowledge  of  the 
meaning  of  the  processes.  So  a  set  of  problems  that  are 
difficult  enough  to  be  a  real  test  of  ability  for  eighth  grade 
pupils  would  be  too  hard  to  be  a  test  for  third  or  fourth 
graders.  Likewise,  a  test  fit  for  children  in  the  lower 
grades  is  too  easy  to  test  the  efficiency  of  upper  grades. 


242  THE   TEACHING   OF   ARITHMETIC 

Test  number  seven  is,  by  its  nature,  the  most  satisfactory 
test  of  the  series  from  an  administrative  standpoint. 
The  test  is  a  general  one  covering  the  written  processes 
in  addition,  subtraction,  multiplication,  and  division. 
But  the  examples  are  not  of  the  same  weight.  In  order 
to  give  work  for  all  grades  from  the  third  to  the  eighth 
inclusive,  the  first  examples  are  very  easy.  Since  the 
scores  of  the  early  grades  are  based  upon  the  very  easy 
examples,  and  since  all  are  reckoned  as  equal  units  in 
making  up  the  scores,  the  growth  from  grade  to  grade  is 
much  greater  than  appears  from  the  scores.  Hence  it  is  not 
a  test  to  be  used  in  estimating  growth  from  grade  to  grade. 

Test  number  eight  is  a  test  in  reasoning  and  computa- 
tion combined.  It  consists  of  two-step  problems.  As  to 
length  of  statement,  all  are  mechanically  of  the  same 
length.  That  is,  there  are  about  the  same  number  of 
printers'  ems  in  the  composition  of  each  of  them.  But 
that  does  not  insure  that  they  are  all  of  equal  units  as  to 
difficulty.  The  degree  of  difficulty  depends  upon  the 
experiences  of  the  pupil.  But  granting  that  the  prob- 
lems are  equal  in  weight,  manifestly,  a  test  in  reasoning 
sufficiently  difficult  to  measure  such  ability  of  a  pupil  in 
the  eighth  grade  is  too  difficult  to  measure  the  power  of  a 
third  or  fourth  grade  pupil. 

THE  TESTS  A  REAL  CONTRIBUTION  TO  EDUCATION 

While  the  work  of  Courtis,  being  the  pioneer  work  in 
this  new  field  of  investigation,  has  its  defects,  and  some  of 
his  early  tests  were  not  well  chosen  for  the  purpose  for 
which  they  were  intended,  he  has  rendered  the  cause  of 


MEASURING   RESULTS  243 

education  a  distinct  service  in  making  the  beginning. 
His  standards,  however,  are  based  upon  the  present  attain- 
ments of  the  schools  examined.  They  should  not  be  taken 
by  school  men  as  an  ideal  of  attainment  toward  which  their 
schools  should  work.  They  may  be  too  high  or  too  low. 
But  surely  a  standard  made  as  these  were  is  much  more 
probably  a  safe  one  than  one  made  from  the  snap  judgment 
of  some  individual,  and  such  a  standard  goal  of  attain- 
ment is  going  to  lead  to  more  uniform  work  throughout 
the  country. 

But  before  we  can  have  much  more  than  a  mere  opinion 
of  what  a  reasonable  standard  should  be,  there  must  be  a 
much  wider  range  of  investigation.  Standards  based  upon 
conditions  as  they  are  in  no  way  show  what  degree  of 
efficiency  is  desired;  that  is,  they  do  not  show  a  most 
economical  use  of  a  pupil's  time.  If  we  can  find  by  tests 
how  much  skill  attained  in  any  grade  is  carried  over  into 
future  school  work  or  out  into  the  various  adult  activities, 
and  the  relative  loss  of  high  or  low  attainments  during  long 
or  short  periods  of  disuse,  and  many  other  such  problems, 
we  shall  then  be  in  a  much  better  position  to  state  a  more 
scientific  standard  of  efficiency.  That  is,  we  will  be  in  a 
much  better  position  to  decide  when  training  in  computa- 
tion, or  any  other  ability,  in  any  grade,  ceases  to  be  an 
economical  use  of  a  pupil's  time. 

There  should  be  established,  carefully  and  scientifically, 
a  measure  known  to  all  teachers,  and  one  that  can  be 
used  by  them  at  any  time  and  without  extra  expense  or 
extra  use  of  time.  Thus,  we  should  be  able  to  say  that 
ability  to  add  twelve  exercises,  each  consisting  of  six 


244  THE   TEACHING  OF  ARITHMETIC 

three-figured  numbers,  in  ten  minutes,  ten  of  the  twelve 
sums  being  correct,  is  ability  of  a  certain  grade.  Such  a 
standard  for  each  grade  and  each  process  would  encourage 
a  teacher  to  work  with  a  definite  aim  in  mind. 

THE  WORK  OF  WOODY 

In  1916  Dr.  Clifford  Woody1  made  an  arithmetic 
scale  of  measurement  differing  from  those  of  Courtis  in 
several  particulars.  Instead  of  all  exercises  being  of  equal 
weight,  they  ranged  from  the  simplest  primary  number 
facts  through  fractions,  decimals,  and  compound  numbers. 
Thus  they  measure  abilities  too  primary  to  be  measured 
by  the  Courtis  tests  and  also  measure  a  wider  range  of 
abilities.  There  are  two  distinct  series,  known  as  "Series 
A"  and  "Series  B."  Twenty  minutes  is  allowed  for  each 
scale  in  Series  A,  and  ten  minutes  is  allowed  for  each  in 
Series  B.  The  author  recommends  that  in  Series  B  all 
tests  be  taken  in  succession. 

The  tentative  standards  set  up  by  the  author  were  based 
upon  a  total  of  about  20,000  test  papers.  There  are  two 
methods  of  scoring  used.  The  methods  and  scores  are 
not  reproduced  here,  for  before  the  standards  of  achieve- 
ment would  be  of  value  to  a  teacher  she  would  have  to 
know  the  conditions  under  which  the  tests  were  given  and 
the  methods  of  computing  the  scores.  For  these  the 
reader  is  referred  to  Dr.  Woody's  monograph  on  the  sub- 
ject. To  show  the  nature  of  the  tests,  however,  Series  A 
is  reproduced  here. 

1  Measurements  of  Some  Achievements  in  Arithmetic,  Teachers 
College,  Columbia  University,  New  York  City,  1916. 


MEASURING  RESULTS 


245 


THE  WOODY  ARITHMETIC  SCALE 

SERIES  A 
ADDITION  SCALE 

When  is  your  next  birthday? How  old  will  you  be?.. 

Are  you  a  boy  or  girl? ....In  what  grade  are  you? 


(1)  (2)          (3) 

2  2  17 

342 

3 


(10)  (11) 

21  32 

33  59 

35  17 


(12) 

43 

1 

2 

13 


45 


(13) 
23 
25 
16 


(5)  (6) 

72  60 

26  37 


(14) 
25+42  = 


(7) 
3+1  = 


(15)  (16) 

100  9 

33  24 

45  12 
201  15 

46  19 


(8) 

2+5+1: 


(9) 
20 
10 
2 
30 
25 


(17)  (18) 

199  2563 

194  1387 

295  4954 

156  2065 


(19) 
$  .75 

1.25 
.49 


(21) 

$8.00 

5.75 

2.33 

4.16 

.94 

6.32 


(22)  (23) 

547          i+J 

197 

685 

678 

456 

393 

152 

240 

152 


(24) 

4.0125 

1.5907 

4.10 

8.673 


(25) 


Sf 

62| 
12} 
37J 


(27) 


(28) 


(29) 


5i 


(30) 


(31) 
113.46 
49.6097 
19.9 
9.87 

.0086 
18.253 
6.04 


(32) 

i+i+i' 


(33) 
.49 
.28 
.63 
.95 

1.69 
.22 
.33 
.36 

1.01 
.56 
.88 
.75 
.58 

1.10 
.18 
.56 


(34)  (35) 

+i  =  2ft.  6  in. 

3  ft.  5  in. 

4  ft.  9  in. 


(36) 

2  yr.  5  mo. 

3  yr.  6  mo. 

4  yr.  9  mo. 

5  yr.  2  mo. 

6  yr.  7  mo. 


(38) 
25.091  +100.4  +25  +98.28  +19.3614  = 


246  THE   TEACHING  OF   ARITHMETIC 

SERIES  A 
SUBTRACTION  SCALE 


Name  



When  is  your  nert  b 
Are  you  a  boy  or  gir 

(1)         (2)         (3) 
862 
501 

irthday? 

How  old  will 

you  be?. 

1  ?  .         .        .     In  wl 

lat  grade  are  you? 

(7)         (8)         1 
13          59 
8          12 

(4)         (5)         (6) 
9            4            11 
347 

:»)      do)      (ID 

78         7  -4  =          76 
37                              60 

(12)           (13)           (14)           (15)         (16) 
27             16             50             21           270 
3               9             25               9            190 

(17)          (18) 
393         1000 
178           537 

(19)               (20) 
567482         2J-1  = 
106493 

(21)                  (22) 
10.00            3i-}  = 
3.49 

(23) 
80836465 
49178036 

(24)             (25) 
8}               27 
5}                12f 

4yd. 
2yd. 

(26) 
1  ft.  6  in. 
2  ft.  3  in. 

(27) 
5  yd.  1  ft.  4  in. 
2  yd.  2  ft.  8  in. 

(28) 
10-6.25  = 

(32) 
1912    6  mo.    8  da. 
1910     7  mo.  15  da. 

(29) 
75f 
52} 

(30) 
9.8063-9.019  = 

(34)                (35) 
6}            81-11- 

(31) 
7.3-3.00081  = 

(33) 

A-&- 

SERIES  A 
MULTIPLICATION   SCALE 


Name  
When  is 
Are  you 

(1) 
3X7- 

(8) 
50 
3 



your  next  birthdi 

iy?    He 

•w  old  will  you  be?  — 

i  ftr«  von  ? 

In  what  grade 

(2) 
5X1  = 

(9)             (10) 
254              623 
6                  7 

(3)                    (4)                  (5) 
2X3=             4X8=              23 
Jt 

(11)               (12)               (13) 
1036              5096              8754 
868 

(6) 
310 

_4 

(14) 
165 
40 

(7) 
7X9  = 

(15) 
235 
23 

(16) 
7898 
9 

(17) 
145 
206 

(18)                (19) 
24                 9.6 
234                    4 

(20) 
287 
.05 

(21) 
24 
2J 

(22) 
8X5|  = 

(23) 
UX8- 

(30) 
2.49 
36 

(35) 
987} 
25 

(24) 
16 

(31) 

iiXJ|  = 

(36) 
3  ft.  5  in. 
5 

(25)                 (26) 
1X1"             9742 
59 

(32) 
6  dollars  49  cents 
8 

(27) 
6.25 
3.2 

(33) 
2|X3i 

(38) 
.0963  i 
.084 

(28) 
.0123 
9.8 

(29) 
JX2- 

(34) 

i+i- 

(39) 
8  ft.  9}  in. 
9 

(37) 
2iX4|XH  = 

MEASURING  RESULTS 


247 


Name 

SERIES  A 
DIVISION  SCALE 

When  is  your  next  bi 
Are  you  a  boy  or  girl 

(1)                    (2) 
375               972T 

rthday? 

B 

.ovr  old  will  you  be?  

!e  are  you  ?  

y 

..In  what  grad 

(3) 

&T 

(4) 
ITS 

(5) 
9735 

(6) 
373T 

(7)                   (8) 
4  -5-2  =              970 

(9) 

17T 

(10)                        (11) 
6X  =30              2713" 

(12) 
2-1-2  = 

(13) 

(14) 
8)5856 

(19) 

248  -f-7  = 

(15) 
1  of  128 

(20) 
2.17253 

(16) 
68)2108 

(21) 
25)9750 

(17) 
50-J-7- 

(22) 
2)13.50 

4)24  Ib.  8  oz. 

(18) 
13755055 

(23) 
23)159" 

(24) 
75)2250c 

BB 

(25) 
2400)504000 

(26) 
1272^76 

(27) 
J  of  624  = 

(28) 
.003)  .0936 

(29) 
3i-S-9  = 

(30) 
J-i-5- 

(31) 
8-1  = 

(32) 
9f-i-3f  = 

(33) 
52)3756 

(34) 
62.50  -J-li- 

(35) 
531)37722 

(36) 

9)69  Ib.  9  os. 

THE  CLEVELAND  SURVEY  TESTS 

A  series  of  fifteen  tests  used  in  the  Cleveland,  Ohio, 
Survey *  of  1915  is  given  here.  As  will  be  seen,  they 
range  from  the  primary  number  facts  to  simple  operations 
with  fractions.  It  was  the  intention  of  the  authors  of  the 
tests  to  make  tests  that  would  show  both  the  complexity 
of  the  processes  which  a  given  grade  can  master  and  also 
the  number  of  examples  of  a  given  type  that  can  be  per- 
formed in  a  given  time. 

1  Cleveland  Education  Survey,  by  C.  H.  Judd.  Published  by 
The  Survey  Committee  of  the  Cleveland  Foundation,  Cleveland, 
Ohio,  1916. 


248 


THE   TEACHING  OF  ARITHMETIC 


The  Cleveland  Survey 
Arithmetic  Tests 

June  1915 
INDIVIDUAL   SCORE   SHEET 


Name Age 

Grade Date  .... 

School....  ....Teacher ... 


SCORE   IN   NUMBER   OF  EXAMPLES  RIGHT 


Set  A 

SetB 

SetC 

SetD 

SetE 

Set  F 

Set  G 

Set  H 

Set  I 

Set  J 

Set  K 

Set  L 

Set  M 

Set  N 

Set  O 

Total  Score  in  Points  for  Whole  Test.. 


INSTRUCTIONS   FOR    CHILDREN 

1.  Obey  promptly  all  signals  from  the  examiner,  who  will  tell 

you  when  to  begin  working  and  when  to  stop. 

2.  Do  all  your  work  directly  on  this  paper.     Work  steadily  and 

rapidly,  but  do  not  hurry.     Only  the  answers  that  are 
right  will  be  counted. 


MEASURING  RESULTS 


249 


SET  A  — Addition  — 


1 

2 

6 
6 

9      0 
5      1 

4 
2 

1 
3 

7 
7 

9      3 
6      0 

2 

4 

1 
5 

3      6 
8      9 

0 

3 

8      9 

7 

8 

2 

1      4 

8 

0 

2      3 

4 

7 

0      3 

1 

2 

5 

6      7 

5 

8 

6      9 

4 
3 

2 
2 

9      7 
3      8 

4 
0 

5 
2 

7 
1 

4      8 
9      6 

0 
0 

3 

4 

9      2 

1      8 

5 

7 

0 
4 

6      2 
3      1 

4 
8 

5 
9 

1 
0 

6      3 
2      3 

7 
4 

9 
8 

0      4 
6      5 

Total 

Number 

Right 

SET 

B  — 

Subtraction  — 

9 
9 

7 
3 

11 
6 

8 
1 

12 
3 

1 
0 

9 

7 

13 
8 

4 
3 

12 
6 

8 
0 

11 
9 

12 
7 

5 
1 

10 
2 

6 
0 

11 
7 

15 
8 

10 
_9 

12 
4 

2 
1^ 

7 
5 

13 

7 

3 
2 

10 
_5 

1 
1 

6 
3 

15 
_9 

4 
2 

8 
3 

4 
4 

10 
7 

13 
5 

10 
1 

9 
4 

5 
6 

8 
6 

17 
9 

6 
4 

11 
JJ 

6 
0 

12 

9 

15 
6 

5 
3 

16 
8 

7 
0 

8 
5 

16 

7 

9 
1 

11 
4 

Total  Number  Right 


250 


THE    TEACHING  OF   ARITHMETIC 


SET  C  —  Multiplication  — 


3 

4 

9 

0 

5 

4 

2 

7 

4 

9 

2 

7 

8 

2 

6 

1 

9 

6 

0 

5 

9 

6 

4 

7 

6 

2 

3 

9 

0 

7 

1 

2 

8 

0 

5 

1 

3 

6 

5 

4 

1 

2 

7 

0 

8 

7 

3 

9 

2 

4 

6 

8 

7 

6 

3 

1 

8 

9 

0 

3^ 

1 

4 

8 

0 

4 

1 

6 

8 

0 

9 

6 

4 

9 

3 

5 

4 

2 

8 

7 

3 

1 

3 

6 

0 

3 

2 

6 

7 

6 

4 

7 

4 

8 

0 

9 

2 

3 

9 

5 

6 

Total  Number  Right 

SET   D  —  Division  — 

3)9         4)32      6)36      2)0        7)28  9)9        3)21 

6)48       1)1         6)10      2)6        4)24  7)63      6)0 

8)32       1)8         5)30      8)72      1)0  9)36      1)7 

2)10       7H2       1)1        6)18      3)6  4)20      7)49 

1)3        2)8        6)6        3)27      8)64  1)2        4)16 

6)0        3)24      9)36      2)4        8)24  7)7        2)18 

6)42      3)0        7)21      4)4        3)16  9)81      7)0 

Total  Number  Right 


MEASURING  RESULTS 


251 


SET  E  —  Addition  — 


5 

2 

9 

2 

6 

1 

4 

9 

2 

8 

8 

8 

3 

4 

6 

7 

2 

8 

0 

6 

4 

2 

5 

1 

0 

6 

7 

0 

8 

5 

3 

5 

4_ 

1^ 

6 

6 

8 

£ 

4 

3 

6 

2 

6 

8 

6 

4 

1 

3 

7 

7 

2 

5 

9 

0 

4 

7 

8 

3 

3 

1 

6 

8 

1 

2 

5 

4 

9 

3 

3 

5 

8 

9 

5 

1 

3 

8 

8 

5 

4 

6 

SET  F  —  Subtraction  — 


Total  Number  Right 


1009 
269 


1248 
709 

1335 
419 

908 
258 

768 
295 


1365 
618 

707 
277 

519 
324 

1269 
772 


1092 
472 

816 
335 

1236 
908 

615 
527 


716 
344 

1157 
908 

1344 
818 

854 
286 


SET  G  —  Multiplication  — 


Total  Number  Right 


2345 
2 

9735 
5 

8642 
9 

6789 
2 

2345 
6 

9735 
9 

2468 
3 

6789 
6 

3579 
3 

2468 
7 

5432 
4 

5432 
8 

9876 
8 

8642 
5 

3579 
7 

9876 
4 

3689 
5 

2457 
6 

9863 
4 

7542 
7 

Total  Number  Right 


252 


THE   TEACHING  OF  ARITHMETIC 


SET   H  —  Fractions  — 

3     1=  ?_1  = 

5+5~  9    9~ 


M- 

9^9 


9     9 
13 

9+9: 


7     1 


5     5 

6_2 
7     1 


8     8 

4_3 

8     8 


SET  I  —  Division  — 


4)55424  7)65982 


-4-1=  8_7 

9     9~  9    9= 

1     4_  6_2 

6     1_  6_6 


9*9  99 

2     1=  ^_*  = 

7+7~  9    9~ 

4     2=  Z_5  = 

it  99 

Total  Number  Right 

2)58748  6)41780 


9)98604  6)57432  3)82689  6)83184 


8)51496  9)75933  8)87856  4)38968 


Total  Number  Right 


10 


MEASURING  RESULTS 


253 


SET   J  — Addition  — 


7  9 

4 

7 

2 

9 

6 

7 

7 

8943 

2 

.0 

1 

5  2 

5 

1 

9 

6 

9 

1 

8 

0631 

1 

fc 

> 

4  4 

8 

9 

4 

2 

6 

5 

5 

7377 

6 

— 

— 

2  8 

1 

4 

8 

4 

7 

1 

4 

1476 

6 

6  2 

4 

3 

5 

7 

0 

4 

1 

8609 

1 

0  7 

8 

2 

1 

1 

4 

6 

8 

6226 

8 

5  5 

5 

8 

5 

3 

3 

5 

2 

1393 

6 

1  3 

1 

5 

2 

9 

7 

3 

1 

3964 

9 

8  6 

3 

2 

4 

2 

1 

3 

3 

7266 

7 

3  1 

9 

7 

3 

3 

6 

7 

9 

4234 

6 

2  4 

6 

7 

6 

8 

0 

6 

8 

9842 

2 

9  8 

3 

1 

7 

6 

6 

1 

4 

4689 

2 

9  8 

6 

9 

6 

5 

6 

7 

6 

4689 

4 

Total  Number  Right 


SET   K  —  Division 


21)441 


32)672 


23)483 


61)1173 


71)1662  42)882  32)992  61)1342 


63)1166  22)462  21)1071  62)1092 


61)1122  41)861  31)961  41)1681 


61)1281  22)484  31)661 


33)693 


Total  Number  Right 


16 


16 


254 


THE   TEACHING  OF  ARITHMETIC 


SET  L  —  Multiplication 

8246     3597 
29       73 

5739     2648     9537 
86       46       92 

|  Number 

§ 
? 

I 

CO 

4268 
37 

7593 
64 

6428     8563     2947 
68      207       63 

Total 

Number  Right 

30 

SET  M  —  Addition— 

7493 
9016 
6487 
7691 
6166 

8937  8625 
6345  4091 
2783  3844 
4883  8697 
1341  7314 

2123 
1679 
6555 
6331 

6808 

5142    3691 
0376    4526 
4955    7479 
9314    2087 
5507    8165 

30 

5226 
2883 
2584 
0058 
2398 

9149  6268 
8467  7726 
0251  8331 
7536  5493 
5223  3918 

9397 
6168 
3732 
4641 
7919 

7337    8243 
2674    6429 
9669    9298 
6114    7404 
8154    2575 

Total 

Number  Right 

SET  N- 

—  Division  — 

67)32763 

48)28464 

97)36084 

69)29382 

78)69888 

88)34496 

69)40296 

38)26562 

Total 

Number  Right 

30 

MEASURING  RESULTS 
SET   O  —  Fractions  — 


3m 
— 1~± 

4^18 


255 


14     4 


35  20     1 

4X6=  2l"6  = 


12^8  6     21  6     20  12  ' 


G.^m~          A  '  IK 
o     10  o     15 

Total  Number  Right 


30 


Instructions  for  Examiners 

Have  the  children  fill  out  the  blanks  at  the  top  of  the  first  page. 
Have  them  start  and  stop  work  together.  Let  there  be  an  inter- 
val of  half  a  minute  between  each  set  of  examples.  The  time  al- 
lowances given  below  must  be  followed  exactly. 

Set  A 30  seconds  Set  F 1  minute  Set  K 2  minutes 

Set  B 30  seconds  Set  G 1  minute  Set  L 3  minutes 

Set  C 30  seconds  Set  H 30  seconds  Set  M 3  minutes 

SetD 30  seconds  Set   1 1  minute  Set  N 3  minutes 

Set  E 30  seconds  Set  J 2  minutes  Set   0 3  minutes 

Have  the  children  exchange  papers.  Read  the  answers  aloud 
and  let  the  children  mark  each  example  that  is  correct,  "C." 
For  each  set  let  them  count  the  number  of  C's  and  write  the 
number  at  the  end  of  the  set  in  the  first  column ;  also  on  the 
first  page. 

The  teacher  should  multiply  the  number  of  examples  right  by 
the  number  in  the  second  column,  writing  the  result  in  the  third 
column.  Then  find  the  sum  of  the  scores  in  the  third  column  for 
a  total  score. 


256 


THE    TEACHING   OF   ARITHMETIC 


STANDARD    SCORES 

MEDIANS   IN    EACH    ARITHMETIC    TEST    FOR    ALL 
GRADES 


TEST 

GRADE 

3 

4 

5 

6 

7 

8 

A 

13.4 

17.8 

22.2 

24.8 

26.7 

27.5 

B 

9.3 

13.4 

17.2 

19.8 

21.5 

26.0 

C 

6.5 

12.0 

15.5 

16.6 

17.7 

19.0 

D 

6.3 

12.4 

15.7 

18.5 

20.8 

22.5 

E 

4.3 

5.3 

6.3 

6.8 

7.5 

7.8 

F 

2.0 

4.9 

6.7 

7.5 

8.6 

10.1 

G 

2.0 

3.9 

5.2 

5.5 

5.9 

6.6 

H 

0.0 

0.0 

5.0 

5.5 

7.7 

•     8.5 

I 

0.6 

1.1 

2.0 

3.1 

4.0 

4.7 

J 

1.9 

3.2 

4.0 

4.4 

4.9 

5.7 

K 

0.0 

4.0 

6.8 

8.5 

10.1 

12.5 

L 

0.0 

1.7 

2.5 

2.8 

3.2 

3.9 

M 

1.4 

2.5 

3.2 

3.8 

4.4 

5.1 

N 

0.0 

0.8 

1.3 

1.7 

2.0 

2.6 

0 

0.0 

0.0 

0.0 

3.1 

4.1 

5.5 

SOME  CAUTIONS  TO  OBSERVE 

It  must  be  observed  by  teachers  and  supervisors  that 
the  "standard  tests"  which  we  have  discussed  are  but 
measures  of  the  mechanical  aspects  of  the  subject  of 
arithmetic.  There  is  but  little  if  any  relation  between  the 
abilities  to  compute  and  the  power  to  interpret  a  problem 
and  reason  out  what  processes  to  apply.  There  is  a  dan- 
ger that  this  may  be  overlooked  by  some  and  that  a 


MEASURING   RESULTS  257 

teacher  may  overemphasize  the  mechanical  side  of  the 
subject,  which  is  so  easily  measured,  to  the  neglect  of  the 
more  important  part  of  it,  for  which  there  can  be  no  such 
definite  tests. 

For  just  as  abilities  to  spell  and  to  write  are  of  no  value 
unless  we  have  thoughts  to  express  through  such  abilities, 
so  abilities  to  compute  are  of  no  value  unless  we  can  inter- 
pret a  problem  and  know  what  processes  to  apply  to  its 
solution. 


Accuracy, 

standards  of,  6,  221. 
Adding  zeros,  36. 
Addition, 

carrying  in,  40. 

checks  in,  43. 

decimals,  118. 

drills  in,  38,  43. 

importance  of  drills,  33. 

order  of  drills,  32. 

primary  facts,  30,  33. 

single  columns,  39. 

use  of  objects  in,  32. 

written  work  in,  39. 
Addition  facts, 

chart  of,  37. 
Aims,  importance  of,  8. 
Analysis  of  problems,  ch.  XIV. 
Applications, 

business,  130. 

of  percentage,  129. 
Approximating  results,  181. 
Areas,  152. 

of  a  circle,  157. 

of  a  parallelogram,  154. 

of  a  rectangle,  153. 

of  a  trapezoid,  156. 

of  a  triangle,  155. 
Arithmetic, 

aims  of,  8. 

essentials  of,  22. 

practical  values,  12. 

traditional  values,  9. 


Banking,  143. 

Business  applications,  130. 

Carrying  in  addition,  40. 
Checking  work,  43. 
Cleveland  survey  tests,  247. 
Colburn,  Warren,  1. 
Commercial  discount,  132. 
Commission,  130. 
Complex  fractions,  18,  115.] 
Compound  interest,  138. 
Counting,  ch.  III. 

rational,  25. 

rote,  25. 

to  20;  to  100,28. 
Course  of  study,  ch.  XVI. 

grammar  grades,  216. 

intermediate  grades,  212. 

primary  grades,  207. 
Courtis,  S.  A.,  232. 
Courtis  tests,  the,  232. 

Decimal  fractions,  ch.  X. 

addition  of,  118. 

applications  of,  122. 

division  of,  120. 

multiplication  of,  119. 

subtraction  of,  118. 
Decimal  notation,  117. 
Denominate  numbers,  ch.  XII. 

fundamental  processes  in,  151. 

reduction  of,  150. 
Discipline,  formal,  11. 
259 


260 


INDEX 


Discount,  131. 

bank,  144. 

commercial,  132. 
Division,  ch.  VII. 

charts  for,  69. 

decimals,  120. 

fractions,  112. 

long,  66. 

measurement,  62. 

partition,  62. 

short,  65. 

tables  of,  61. 

written,  65. 

Doctrine  of  formal  discipline,  11. 
Drill  work,  3. 

importance  of,  33. 

motivation  of,  165. 

repetition  of,  194. 

Economy  in  teaching,  1. 
Educational  principles,  3. 
Efficiency  in  computation,  13. 
Elimination  of  topics,  17. 
Estimating  results,  182. 
Exchange,  21. 

Figures,  27. 

Forecasting  results,  182. 
Formal  discipline,  11. 
Fractions,  common,  ch.  IX. 

adding,  103. 

complex,  115. 

division  of,  111. 

early  work  in,  95. 

expressing  a  ratio,  102. 

fundamental  processes,  97. 

ideas  involved  in,  99. 

improper,  106. 

indicated  division,  101. 


Fractions  —  Continued 

multiplication  of,  110. 

notation  of,  96. 

objective  presentation  of,  98. 

part  of  a  whole,  100. 

reduction  of,  106. 

reduction  to  decimal,  121. 

subtraction,  108. 

unit  of,  63. 
Fractions,  decimal, 

(See  Decimals). 

Games,  ch.  VIII. 

"make  believe,"  74. 

motor  activity,  83. 

purpose  of,  70. 

school  room,  71. 

scoring,  71. 
Grades,  work  of, 

eighth,  218.  , 

fifth,  213. 

first,  207. 

fourth,  211. 

second,  208. 

seventh,  217. 

sixth,  214. 

third,  210. 
Grammar  grades, 

work  of,  216. 
Graphs,  159. 
Greatest  Common  Divisor,  17. 

Habit  of  seeing  relations,  170. 
Habituation,  5. 

Insurance,  139. 

fire,  139. 

life,  141. 
Interest,  137. 

compound,  138. 


INDEX 


261 


Interest  —  Continued 

methods  of  computing,  137. 

simple,  134. 
Intermediate  grades, 

work  of,  212. 
Inverse  problems,  20. 
Inverting  divisor,  113. 

Labeling  steps,  189. 

Least  Common  Denominator,  105. 

Least  Common  Multiple,  17. 

Lessons, 

deductive,  201. 

drill,  192. 

inductive,  196. 

three  types  of,  191. 
Lesson,  planning  a,  ch.  XV. 
Lesson  plan,  202. 
Loss  and  profit,  133. 

Measurement,  division,  62. 
Mensuration,  152. 

circles,  157. 

parallelograms,  154. 

rectangles,  163. 

solids,  158. 

trapezoids,  156. 

triangles,  155. 

Measuring  results,  ch.  XVII. 
Metric  system,  22. 
Mixed  numbers, 

adding,  107. 

dividing,  114. 

multiplying,  111. 

subtracting,  109. 
Motivation,  45,  165. 
Multiplication, 

decimals,  119. 

development  of  tables,  52,  56. 


Multiplication  —  Continued 
drills  in,  57. 
fractions,  110.  fc 
mixed  numbers,  111. 
written,  58. 

Number, 

begun  too  early,  1. 
child's  need  of,  2. 

Objective  teaching,  32. 
Oral  arithmetic,  23. 

Parallelograms,  154. 
Partial  payments,  20. 
Partition,  division,  62. 
Partnership,  21. 
Per  Cent, 

ability  to  express,  129. 

finding  a,  124. 

notation  of,  123. 

notion  of,  123. 
Percentage,  ch.  XI. 

applications  of,  129. 

three  problems  of,  124. 
Pestalozzi,  1. 

Planning  a  lesson,  ch.  XV. 
Power  to  see  relations,  168. 
Primary  grades, 

problems  of,  163. 

work  of,  207. 

Principles  of  teaching,  3,  175. 
Problems, 

analysis  and  solution  of,  ch. 
XIV. 

book  vs.  local,  178. 

clarifying  a  process,  45,  164. 

concrete,  176. 

contribute  to  social  insight,  172. 


262 


INDEX 


Problems  —  Continued 

failure  to  solve,  179. 

grammar  grade,  167. 

motivating  drills,  165. 

nature  and  use  of,  45. 

primary  grades,  163. 

purpose  of,  163. 

solution  of,  187. 

wording  of,  166. 
Profit  and  loss,  133. 
Pythagorean  Theorem,  21. 

Quantitative  relations,  16. 

Ratio,  102. 

Rationalization,  5. 

Reading  and  writing  numbers,  29. 

Rectangle,  153. 

Reduction, 

of  denominate  numbers,  150. 

of  fractions,  106. 
Rice,  J.M.,  223. 
Root,  square,  21. 

Social  insight,  15,  172. 
Speed  and  accuracy,  6. 
Square  root,  21. 
Stock  investments,  145. 
Stone,  C.  W.,  228. 


Stone's  tests,  228. 
Subject  matter, 

elimination  of,  17. 

essentials,  22. 

grades,  207. 
Subtraction, 

addition  method,  47. 

decimals,  118. 

fractions,  108. 

meaning  of,  47. 

taking  away  method,  50. 
Standard  tests,  221. 
Standards  of  accuracy,  6. 
Statistics  represented, 

by  graphs,  159. 

Taxes,  142. 

Unit  fractions,  63. 

Values  of  arithmetic, 

practical,  12. 

traditional,  9. 
Volumes,  158. 

Weights  and  measures,  149. 
Woody,  C.,  244. 
work  of,  244. 
Woody's  tests,  245. 
Work  by  grades,  207. 


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